Bike fork

Obviously the fork dropouts need to be deep and wide enough to accommodate the right wheel. The arms should have the right amount of rake for the frame. 

Rake

Bike forks are designed with something called rake. Basically the forks are curved forward to counteract the drag received while riding. This aids the bike in keeping sharp steering as the forks keep the wheel pushing ahead. It is measured as the inline difference between the center of the steerer tube, and the center of the skewer hole in the dropout.

Most road racing bikes have a rake offset of 40-45mm. Large wheels often require a larger rake offset.

Steerer Tube dimensions

Steerer tubes can be different lengths to meet desired handlebar heights on both threaded and threadless headsets. They can't stand higher than the headset stack, and are therefore cut to the proper height upon installation. There are numerous adapter kits for smaller diameter steerer tubes fitting into different headsets. 

A major aspect in mountain bike forks is the suspension. Telescopic, shock absorbing forks emerged in the early 90s, and technology has improved to today's space-age, ultra-sensitive suspensions systems we know and love. Suspension forks usually consist of a spring and a damper. The spring can be any mechanism (metal coil or even compressed air) that provides a varying level of stability to the fork. The damper could be a series of valves which oil passage is limited, providing the suspension with a softer bottom-end during high stress impacts. 

Some relevant suspension terms:

1. Travel: the distance the axle moves
2. Preload: initial tension on suspension springs before external loads are applied. Soft suspension has a less preload, and very rigid suspension has a high preload. 
3. Rebound: refers to the speed at which the suspension returns to its initial state after an external shock. Rebound dampers vary the rebound speed, possibly causing the shock to return to normal more slowly.
4. Sag: The resting height of the suspension with the rider included. This is taken into consideration for adjustment.
5. Lockout: Toggle-like mechanism that bypasses suspension in desired situations that require a more rigid frame. Some lockout systems have a cancelation feature that activates under sudden unexpectedly high loads (like crashing) to protect the bike and rider.   
6. Bob and squat: Phenomenon that occurs when the suspension responds to the pedaling of the rider, robbing some acceleration power.
7. Pedal feedback: With the rear axle moving (and therefore changing the angle in relation to the bottom bracket), the chain is tightened and shortened. This is felt as torque that is opposite to pedaling when riding.
8. Compression dampening: slowing down the rate of shock compression by forcing fluid (air or oil) through a valve. Resistance at the valve vary, and can sometimes be adjusted, therefore softening or hardening the ride. 
9. Unsprung mass: Refers to the collective body of components unsupported by suspension. In the case of mountain bikes, this means only the wheels and other minimal parts below the shocks, but road cycles having no suspension system are nearly 100% unsprung mass. When the rider lifts from the seat, his/her knees provide shock absorption therefore rendering them sprung mass.