There is a saying 'You can't kill what you can't see'. This isn't strictly true, but is more readable than 'If you can't see something then the chances of you being able to kill it are reduced to random chance'.
While this entry mainly refers to aircraft, bear in mind that stealth is equally relevant to land and seabound vehicles, the Sea Shadow being one such example. Stealth is not necessarily about making an object impossible to detect; the idea is to make an object indistinguishable as a threat. For example, the F-117A is believed to have a radar cross section equivalent to a sparrow, so while it could be detected, short of shooting down every single sparrow-sized object, there is not much an enemy can do.
There are different ways of detecting an enemy and hence different methods of making an object stealthy.
The simplest way to detect an aircraft is the Mk1 eyeball. Designers try to reduce the visibility of an aircraft by choosing an appropriate colour scheme. Painting an aircraft is not a quick process, so designers will choose one colour scheme based on the operating environment and the perceived risk. Low-level bombers are painted green on top and grey underneath. To an enemy aircraft above, it will blend in against the ground, while high level bombers and fighter aircraft tend to be painted grey to blend in with clouds. When flying at night, aircraft can also be located by the glow from their engines, so the exhaust has to be designed to cover this.
Any form of combustion such as a car or jet engine will generate heat and will thus appear hotter than its background. This can be detected by an enemy and used to guide missiles against the aircraft. To counter this, the engine is designed to produce as little heat as possible, which normally means not much thrust is produced. However, the exhaust is still going to be several hundred degrees warmer than the surrounding atmosphere. To disguise the excess heat, the exhaust of the engine contains several baffles (a baffle is basically a big can). The idea is that these baffles mix in cold air from the atmosphere to reduce the temperature of the air flowing out of the exhaust to undetectable levels.
This is the main method employed for locating aircraft, for both civilian and military purposes, and involves bouncing a radio signal off an object, which can then be detected by a receiver. Stealth aircraft defeat this in two different ways. Firstly, surfaces on the aircraft are constructed out of radar absorbent or transparent material. Secondly, structures within the aircraft are designed to minimize the size of flat areas, which would reflect a signal back at the radar. The structure is built with lots of different facets. When a radar beam hits the aircraft, the radar's energy is deflected away at many different angles so very little of the radar energy is reflected back at the radar receiver.
It is rumoured that the US are developing an active method of defeating enemy radar. This involves detecting the frequency of a radar pulse and then sending back a signal that is exactly 180° out of phase. This has the effect of cancelling the signal out so the radar is unable to detect a return signal.
While not an ideal method for locating an opponent, the fact an enemy knows you are there puts you at a disadvantage. From an operational point of view, a pilot can reduce this risk by flying at sub-sonic speeds. Also, as with the heat signature, baffles on the engine can be used to reduce the noise generated by the engines.
The whole point of being stealthy is so that you can approach a target undetected and destroy it. The fact that the building you are guarding has just blown up for some unknown reason may indicate there is a stealth aircraft out there. An enemy could start firing, relying on random chance to hit the aircraft. To avoid this slim possibility, designers have developed stand-off weapons so that the aircraft can stay out of range. The aircraft will orbit the target at a safe distance and then release some form of missile. Depending on the altitude this may glide to the target or have some form of propulsion. It is then guided in by remote control.
I've Built my Stealth Aircraft - What Do I Do with It?
In order to achieve stealth, the designers have to make a number of design compromises. Truly stealthy vehicles normally suffer from severe performance limitations. The F-117a is allegedly incapable of flying at supersonic speeds, has a very limited payload, and manoeuvres like a brick. However, the ability to sneak up on an enemy and kill them before they even know you are there more than outweighs these disadvantages. The operational concept is that stealth aircraft would be used to take out radar defences and communications centres, in effect blinding the enemy so that more conventional aircraft can follow in their wake.