How does the Space Shuttle slow down during re-entry, descent, and landing?
If you're interested in a more visual, and less technical, explanation of Space Shuttle reentry and landing, I gave a talk titled How to Land the Space Shuttle... from Space at the Stack Overflow meetup in October 2016.


I didn't notice this question until a couple of days ago, but as someone with an unhealthy obsession with specifically the entry and landing phases of shuttle flights, I can say there is a lot of factually incorrect information in the other answers here. Let me see if I can explain it better.

First, the two easy questions, which were answered well by other questions, but I'll include here as well for completeness:


Could the shuttle perform a go-around? No. The OMS engines are too weak to make a difference in the atmosphere, and the main engines (which would be powerful enough) are only fueled by the orange external tank which is jettisoned after launch.
Where did it land? 78 missions landed at the Kennedy Space Center, 54 (including the first) at Edwards Air Force Base, and 1 at White Sands. There were other landing sites designated for emergencies, but none were ever used.


Now, for the really big question of how the shuttle reentered and landed.


  The primary source I'm going to cite in this answer is the Entry, TAEM, and Approach/Landing Guidance Workbook 21002 which was a workbook used for training astronauts. Sadly, I don't have a link to it, but it can be obtained from nasaspaceflight.com via an L2 subscription if you're really interested. I'm going to abreviate this source as ETAGW.


Orbital Mechanics

First, a real quick lesson in orbital mechanics. In order to change the altitude of your orbit you make a change in velocity ($\Delta v$). If you increase velocity, you'll increase altitude, and if you decrease velocity, you'll decrease altitude. However, this effect is most pronounced 180° from the position where you made the change. After a complete 360° orbit, you'll be at approximately the same altitude you started at.

This illustrates the effect:



Starting from the circular orbit (black), if you slow down at point A, you might end up with something like the red orbit, and if you speed up you might end up with something like the blue orbit.

Deorbit Burn

Because of the nature of orbital mechanics, as described above, you want to perform your deorbit maneuver on the opposite side of the planet from your intended landing site. This typically occurred over the Indian Ocean for a landing at Kennedy Space Center in Florida.

The burn itself was performed with the shuttle flying tail first and white (top) side facing towards the Earth (heat shield/black side facing towards space). The two OMS (orbital maneuvering system) engines were used to accomplish the required $\Delta v$ (anywhere between 200 and 550 ft/sec depending on the starting altitude). The burn typically took around 2.5 to 3 minutes. This would lower the perigee (lowest point in an orbit around Earth) to within a few miles of the ground (having a hard time finding a source, but I seem to remember it being around 30-40 miles), which was enough to ensure the orbital path would take them into thick atmosphere.

 Image Source

The OMS engines are essentially larger versions of RCS (reaction control system) jets. RCS jets were used for rotation (attitude) and small translation (velocity in a given direction) changes. The OMS were used for making orbital changes.

Both systems burned the same hypergolic mixture (monomethylhydrazine (MMH) and dinitrogen tetroxide (N2O4)). If the OMS had failed, the RCS jets could have, in theory, been used to slow the shuttle enough for reentry.

Nose Flip

Some answers have said that the orbiter would pitch down 140° for the flip. This is incorrect. After the deorbit maneuver was completed, the orbiter would pitch up about 220° until it reached a 40° nose-up angle of attack (referred to as the "EI -5 attitude" because they must be in that attitude by at least five minutes prior to entry interface).

They have about 20 minutes between the deorbit burn and EI-5, so there's plenty of time to pitch in either direction. Nose up was likely preferred because the APU exhaust vents point up near the tail. This naturally causes the orbiter to want to pitch up when the APUs are running.

During this pitch-up, they would fire the forward RCS jets to dump all the forward fuel, unless it was required for center-of-gravity reasons. The forward RCS jets weren't used during reentry, and dumping the fuel reduces potential hazards to the astronauts.

Once in position, open-loop entry guidance would begin, holding the orbiter at 0° roll (wings level), 0° yaw, and 40° angle of attack (alpha).

Entry Interface

NASA defines the entry-interface (EI) as an altitude of 400,000 feet. There's no hard-edge to the atmosphere, but this is around the altitude where its effects start to become directly detectable.

S-Turns

Some answers have claimed that the shuttle used S-turns for the purpose of slowing down. This is a very common, but over-simplified and arguably inaccurate, explanation.


  "The next time you hear someone talk about the shuttle doing roll reversals to bleed off energy, do not listen. The shuttle does roll reversals because it has a very small alpha envelope." - ETAGW 2.8.1 (emphasis is from original source)


As with any airplane, the shuttle's wings generate lift. As the atmosphere gets denser, the wings are going to generate more lift, and this upward lift will cause the descent rate to slow. In fact if the shuttle maintained a wings-level attitude, it would eventually start to gain altitude causing it to "skip" across the atmosphere several times until it was slow enough to fall through. And while technically a skipping reentry would be possible, it would be very difficult to control with any precision.

So, instead, when the decent rate starts to slow, the shuttle goes into a bank. By controlling the bank angle, they're able to control how much upward lift the wings are generating, and, by extension, control their descent rate.


ETAGW Figure 2-5 illustrating lift vector.

Or course, with the lift vector pointed sideways instead of upwards, the shuttle is going to start to turn. Due to the incredible speed, the turn radius is enormous, but it does gradually turn nonetheless, and the orbiter develops an azimuth error (the difference between the orbiter's current direction and the direction to the landing site).


ETAGW Figure 2-3 illustrating azimuth error.

To correct this azimuth error ($\Delta z$), the orbiter performs "roll reversals". In other words, it turns in the other direction. These turns create the distinctive S-turn reentry track.

The first reversal always occurs at 10.5° $\Delta z$. Subsequent reversals occur at 17.5° until Mach 4 when it starts ramping down to 10° at Mach 3.


  Note: all mach numbers given anywhere in this answer, or referenced anywhere in Space Shuttle materials, aren't true mach numbers. NASA uses 1000 ft/sec as an approximation of Mach 1, and all mach numbers are multiples of that velocity.


Ranging

Obviously, the ultimate goal of reentry is to reach your intended runway at an appropriate speed for landing. While the orbiter doesn't have any engines to help accomplish this, it does have a tremendous amount of orbital energy. Therefore the goal becomes energy management, and more specifically, drag management.

The orbiter has two ways to affect drag during entry: changing angle of attack (alpha), and changing bank angle.

Angle of attack is the quickest way to increase or decrease drag, but the orbiter was only allowed to deviate ±3° from the nominal alpha (40° for most of entry). This is the "very small alpha envelope" referred to earlier. The limitation is designed to ensure proper heat protection and maintain vehicle control.

Changing bank angle, as described above, allows you to control your descent rate. A steeper bank angle will result in reaching thicker air faster, and will therefore result in increased drag. A shallower bank angle will keep the orbiter in thinner air for longer, and result in less drag. However, it takes a little longer to see the effect of a bank change than an alpha change.

There are also bank angle limitations because, again, you don't want to skip out of the atmosphere, and you don't want to fall into thick air so fast that it exceeds the maximum drag the vehicle can handle, but it is a larger envelope than the alpha limits. In fact, in early stages of entry, the orbiter could have even flown upside down (with its lift vector pointed towards Earth) if necessary due to an under-burn as the result of a malfunction during the deorbit burn.

Entry Guidance Phases

I'm not going to go into detail about entry guidance, but I will say that the primary considerations change as speed and altitude changes, and the entry guidance is broken up into phases to reflect this.


ETAGW

In the chart above, the middle line represents the nominal profile. The lowest line is the "equilibrium glide" profile which is the minimum amount of drag that the orbiter must maintain to avoid gaining altitude and skipping. The line at the top left represents a thermal limit (if the orbiter exceeds it, it might burn up). The line at the upper right represents a dynamic pressure limit (if the orbiter exceeds it, it might break up due to aerodynamic forces).


Preentry: This is considered open-loop guidance because all it does it maintain 0° yaw, 0° roll, and 40° alpha until the total load factor becomes 0.132g (approximately 3 ft/sec2 of drag), at which point closed-loop guidance begins.
Temperature Control: Begins at closed loop guidance and ends at a velocity of Mach 19. Tries to maintain a constant temperature within the design limits of the orbiter.
Equilibrium Glide: Simply provides a bridge between temperature control and constant drag phases. It ends when drag reaches 33 ft/sec2. It's named as such because its shape is similar to that of the equilibrium glide profile.
Constant Drag: Maintains a constant drag rate of 33 ft/sec2 until velocity reaches Mach 10.5.
Transition: Designed to transition from the high drag and high alpha of entry to the lower drag and lower alpha required for the orbiter to fly more like an airplane. This phase terminates at Mach 2.5 when TAEM begins.


TAEM

TAEM stands for Terminal Area Energy Management. The objective in this stage is to get the orbiter lined up with the runway with the correct amount of energy to make its final approach. Again, I'm not going to go into a ton of detail (feel free to ask other questions if you want), but here's the gist:

If all went well in the entry stage, TAEM will begin at about 82,000 feet and 60 nautical miles from the runway (intended ground track, not straight-line distance).

Whereas Entry guidance primarily uses bank angle to manage energy, TAEM primarily uses angle of attack. Below Mach 1, the speedbrake (a split rudder) also helps with energy management.


Source: NASA

Phases of TAEM:


S-Turn: Usually not required, but the shuttle will perform S-turns if it is too high on energy at the start of TAEM (too high, or too close to the runway).
Acquisition: Turns the orbiter towards a point of tangency on the heading alignment cone (HAC) and then flys wings-level until it intercepts the HAC. The tangency point is referred to as "waypoint 1" (WP1). During this phase, the orbiter slows to below Mach 1, at which point the commander takes CSS (control-stick steering), which is the closest thing the shuttle has to a "manual" mode.
Heading Alignment: Guides the orbiter around a virtual "cone" (see diagram below) until it is in alignment with the runway. It's not really a cone, mathematically speaking, but it's the easiest way to visualize it.
Prefinal: Establishes the orbiter on the outer glide slope.



ETAGW Figure 3-13 illustrating the HAC.

Approach and Landing

The final guidance phase is called "Approach and Landing". It begins when the orbiter is below 10,000 feet and established on the outer glide slope (OGS), but no later than 5000 feet regardless of glide slope.

The OGS was a 20° glide slope for "light" orbiters (gross weight less than 222,000 pounds) or 18° for "heavy" orbiters (by comparison, normal airplanes use a 3° glide slope). Heavy or light depended on what was in the payload bay. The nominal aim point for the OGS was 7500 feet short of the runway threshold, but there was also a "close-in" aim point at 6500 feet which was used in the event of a strong enough headwind.

The speed brake was used to maintain 300 KEAS (knots equivalent airspeed - it's effectively the same as indicated airspeed) on the OGS until 3000 feet, at which time it calculated how much speed brake should be required for landing, and moved to that position. It would re-calculate once more at 500 feet.

At 2000 feet, the orbiter would begin a "preflare" maneuver designed to transition from the OGS to the shallow 1.5° inner glide slope (IGS). This was accomplished with a circular pull-up followed by an exponential decay onto the IGS.


ETAGW Figure 4-8 illustrating the preflare geometry.

In reality, the IGS is not followed for long, and is more of a guide for making sure you cross the runway threshold on the proper trajectory and get into position to begin the final flare, which is essentially the same as a conventional landing flare in a normal aircraft, except that the shuttle lands at a much higher angle of attack (about 8°) due to its delta wings (more similar to Concorde).

Landing gear was deployed at 300 feet, and if you're interested in landing gear, I wrote a whole answer on landing gear deployment once.

The targeted touchdown point was 2500 feet down the runway at an airspeed of 195 knots for light orbiters, or 205 knots for heavy (within +5/-10 knots).

The drag chute (an addition made in the early 90's) would be deployed shortly after main gear touchdown, but not faster than 195 knots, and sometimes they would wait until nose gear touchdown if there was a crosswind. It would be jettisoned at 60 knots to ensure the chute attachment mechanism didn't hit and damage the main engines.

Control

(e.g. how was the shuttle able to make attitude changes?)

In early entry, the orbiter is still controlled like a spacecraft, using RCS jets to control attitude. As dynamic pressure (q-bar) increases, the aerodynamic surfaces begin to become active and the RCS jets shutdown as follows:


q-bar = 0.5 pounds/ft2 (psf), the elevons begin to act as trim.
q-bar = 2 psf, elevons begin to act as active control surfaces.
q-bar = 10 psf, RCS roll jets are disabled.
q-bar = 40 psf, RCS pitch jets are disabled.
Mach 10, the speedbrake opens on a pre-programmed schedule to act as pitch trim.
Mach 5, rudder becomes active, initially acting primarily as aileron trim.
Mach 1, RCS yaw jets are disabled.


Navigation

(e.g. how did the shuttle know where it was at?)

Navigation, in NASA terms, basically means knowing where you are (guidance answers the question "how do I get where I want to go?"). During entry, the shuttle primarily used inertial navigation units, which were aligned using a star tracker prior to the deorbit burn. As it got closer to the landing site it could also incorporate GPS and/or TACAN signals into the nav data. OV-105 (aka Space Shuttle Endeavor) had a three GPS units installed and no TACANs, whereas the other orbiters had three TACANs and one GPS unit.

On the HAC, anywhere from about 15k to 20k feet in altitude, the orbiter would pick up the microwave landing system (MLS), which acts sort of like a very high precision ILS. From this, they're able to determine their position with high precision and accuracy. Once acquired, this becomes the primary source of nav data for the remainder of the flight.

On landing videos, you can actually see exactly when they pick up MLS because the altitude tape goes from being sort of jerky, then jumps a little bit and becomes smooth. For example, watch here at about the 20k mark (altitude tape is the one on the right).

Below 5000 feet, the orbiter was also able to use a radar altimeter for altitude information.



That's probably more information than you were looking for, but if you have questions about things I didn't cover, or didn't go into enough detail on, feel free to ask separate questions for those.