Will I get electrocuted by touching an exposed wire of a closed circuit?
You've got good answers as far as some of the safety aspects, so I'm just going to clarify something about the ground.  There is an additional benefit to the low impedance path it provides.  The point is not simply for more current to flow through the ground and less through you, the point is to have so much current flow through the low impedance path provided that the circuit breaker or fuse trips nearly instantly.  This greatly decreases the danger of faults since for you to be shocked as an additional path to ground, you have to be in contact with the device in question at the moment of the fault.  The scaffold incident mentioned in another answer shows where metal is providing a path to ground but people are shocked anyways, but I would note the scaffold has rubber wheels, which even when wet, would isolate the upper metal part of the scaffold.  The wheels also have low surface area in contact with pavement, which is itself a poor conductor, even when wet. If they had had a ground conductor terminated to the scaffold and the metal parts of the scaffold bonded together, it is possible the current would have mostly bypassed the workers, leaving them alive.
An electrical danger chart from wikipedia:

It's for AC current and on the vertical is the time you're exposed and the horizontal is amount of current so we can look and see that 50mA for a second or more may stop the heart, and any current above that exceeds what is necessary to kill.  120 volts is enough voltage with the average person in average clothing.  What exactly constitutes a &quot;Low impedance ground path&quot; is affected by the source voltage, impedance and current limit.  In the case of 120v or 240v residential lines, a line protected by a 50A circuit breaker requires a larger grounding/bonding wire than one protected by a 15A circuit breaker because for the breaker to trip, a fault at the same voltage must cause more than 3 times as much current to flow.  The faster the surge, and the faster the breaker trips, the shorter the surge is.  You may have noted that &quot;low impedance path to ground&quot; is quoted, not &quot;low resistance&quot;.  The ground network must not just be low enough resistance to allow a high current flow and trip the breaker, it must also be low inductance to prevent delay in tripping due to inductive effects.
Now let's look at how much resistance a human has:

You can see from the chart on the right that if you're wet you have much lower resistance, so lets say you're in the rain and you're leaning with one hand on the casing of a piece of equipment.  Wet palm touch says your resistance is as low as 1000 ohms.  If that piece of equipment is ungrounded and becomes energised to 120VAC, current flow will be equal to
I=E/R
I=120V/1000Ω
I=120mA
So that has a good chance of being lethal.  Let's say the device is grounded at the time of the short such that 100 amps flow for a moment on a 15A line, tripping the circuit breaker almost instantly(100A is conservative), in the instant the short occurs and that current flows, the resistance of the 15A supply line and source impedance will cause a voltage drop, end effect being that you are exposed to much less than 120V for less than a hundredth of a second.
Or to look at a piece of equipment fed by a 75f run of 14AWG copper wire, the electricity has to go both ways, so we can calculate the resistance of 150f of 14 gauge wire as 0.379 ohms.  So if we apply a perfect 120V to it when the equipment is drawing the maximum 15A, the wires have a volt drop of
E=IR=15A*0.379Ω=5.685V
So even with the full 15A rated current flowing and a reasonable length wire run you can get significant voltage drop.  You only have 114.315V at the device.  Now lets assume a 14 gauge ground wire and look at how much current we would expect in this example with a short.
I=E/R=120V/0.379Ω=316 amps
Needless to say this will cause a significant voltage drop that will protect you while the current briefly flows, and much more than the roughly ~5x rated current required to trip the circuit breaker near instantly.