My short answer:
- Stability is reduced by shifting the center of gravity aft.
- Shifting it past the neutral point makes the airplane unstable, so movements away from the trimmed state are accelerated. This increases maneuverability.
- Flight computers are multiple redundant, if one dies the others take over.
- Slow unstable airplanes can be flown by a human pilot, but not fast unstable airplanes.
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 got 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 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.
Both configurations have the same airfield and combat performance!
Can a human still fly such an aircraft? In glider competitions, the more daring pilots fly with relaxed static stability and have no problem to keep the aircraft under control. Even the Wright brothers could handle their unstable airplane, and handling improved when they moved the CG further back (If you want to know why, please post a new question. This answer is getting too long already!). However, the speed of an aircraft's pitch response is proportional to flight speed (and inverse to the pitch moment of inertia), so faster aircraft are harder to control. You can compare dynamic pressure with the stiffness of a spring: A stiffer spring moves the eigenfrequency of a spring-mass system up, and the same is true for the eigenvalues of the equations of motion of an aircraft. Given that the reaction time of a good pilot is at least 0.1 s (and more is he/she is tired), it is impossible to counteract motions with frequencies of more than a few Hertz. The lag means that the reaction comes too late and will support the motion. See this YouTube clip how that works out in practice. This crash was due to wrong signal gains, not a classic instability (after all, the flight computer was still working, but produced too strong elevator deflections).
I would venture to say that a human can still barely fly an unstable jet at low speed (after all, Tom Morgenfeld almost got the YF-22 under control), but once he firewalls the throttle, he will be always behind the plane, and will crash it soon.
Size helps: Bigger planes have lower eigenfrequencies, and light, big vehicles are easy to control, regardless of stability. All Zeppelins were completely unstable in yaw and below the critical airspeed of an airship (again, please ask for a more detailed answer on this aspect) also in pitch, but with one person each for the vertical and the horizontal control surfaces, and enough people on board to rotate them after 2 - 4 hours, nobody felt the need to make Zeppelins naturally stable.
If one computer dies, the others take over. Most unstable configurations have four parallel computers which cross-check their result to catch any malfunction. The Dassault Rafale uses only three, but adds safety by clever algorithms for checking the results.