How is an aircraft designed to be longitudinally stable?
My short answer:

Stability is controlled by moving the center of gravity (CoG).
The handbook should give center of gravity limits. A more forward CoG means a more stable airplane.
Shifting it past the neutral point makes the airplane unstable, so movements away from the trimmed state are accelerated. However, at supersonic speed this airplane will become stable again.
Airplanes with sophisticated computer control can artificially augment stability, so they behave like a stable airplane at all speeds.

For the long answer, let me first clarify terms:
Static stability is the tendency of a system to return to it's old state after being disturbed. Take a pendulum: If you pull it to one side, it will return to the middle. Eventually.
Dynamic stability is the tendency of an oscillating system for the oscillations to die down over time. Take the same pendulum: It will swing from side to side, and  friction will ensure this happens with ever smaller amplitude.
Now we need to add dimensions, all three of them: Pitch, roll and yaw. An airplane can be stable in one dimension and unstable in a different one. I understand your question such that you ask about the static pitch stability (or longitudinal stability) of fighter aircraft.
The Wright Flyer was longitudinally unstable (see here for more). Once aircraft designers learned that aircraft can be made to fly stable, and learned that this is of immense benefit in pilot training, static stability became a requirement for new aircraft. When war in Europe broke out, the British forces were equipped with a superb training aircraft, but it was so stable that it took effort and time to convince it to change course. They were shot down in droves.
From now on, low stability was a prime requirement for fighters and aerobatic aircraft. Static stability is proportional to the control forces (more precisely: To the hinge moment of the respective control surface), so reducing stability gave pilots more response for the same effort. Longitudinal static stability is measured as the relative distance between neutral point (NP) and the center of gravity (CG). See here for more. Longitudinal static stability is achieved by placing the CG ahead of the NP. Shifting the CG back gives you a more responsive airplane, but also one which is more easily disturbed by gusts.
This is the design technique you asked about. Pretty simple, right?
Once you shift the CG aft of the NP, stability is lost and the airplane will increase deviations from the trimmed state. This can be helpful if you want large angle changes, and quickly. An unstable aircraft only needs a small kick and will do the rest of the maneuver all by itself.
This is how it helps in maneuverability. But it is even more helpful to reduce the inertias, especially around the roll axis, for a faster response. That is why all combat aircraft have their engines close to the center.
Of course, negative stability is not acceptable when you need to take your hands off the stick to get a map out or to pee on a long flight. So without computer control, the limit was a CG position near, but not aft of the NP.
With supersonic aircraft, things get more complicated. Now the aircraft operates in two flight regimes, one where lift acts at the quarter chord of the wing and one where it acts at mid-chord. Aircraft with low static stability become very stable in supersonic flight, and the tail surface has to create a high down-force so that the sum of all lift stays where the CG is. Creating lift always incurs a drag penalty, and in supersonic flight it has to be payed twice: One for the excess lift on the wing (which is needed to compensate for the tail's down-force) and one for the down-force on the tail.
Using a flight control computer offers the possibility to allow the pilot to let go of the stick without the aircraft going off course. Now the stick does not command elevator deflection, but pitch rate, and the CG can be moved back from maybe 12% of MAC (mean aerodynamic chord) to -2%. If you compare the wing areas of stable and unstable jets (Jaguar and Mirage F-1 are prime examples), you will see how much is achieved just by going back with the CG by a few percent of wing chord.

Conventional SEPECAT Jaguar and CCV version (picture from Ray Whitford's Fundamentals of Fighter Design). Both configurations have the same airfield and combat performance!
These airplanes are unstable without computer control at subsonic speed. This is no benefit in itself (for better maneuverabiliy it helps more to reduce inertias, as explained above), but it dramatically cuts trim drag at supersonic speed. In supersonic flight such an aircraft is still stable, but much less than one that is also stable at subsonic speed. This means less lift is needed on the wings and less downforce is created on the tail, so the drag of botch surfaces is small and their size can be reduced, too, resulting in a virtuous circle of drag savings.