Fortunately, most units follow the same size standards (DIN).   In many cars, once the factory radio is removed the aftermarket radio will fit in the hole.  In many other cars, a kit is needed if the factory hole is too big, or not deep enough.  In some cases the dash has to be cut.  Any car stereo store should have kits required for installation.

Even though not necessary, it is recommended to use a wiring harness when installing an aftermarket radio.  The harness is wired up to the radio, an plugs directly in the factory plug, making a good and easy connection.  Since the factory plugs are not cut, the manufacturer's warranty is not voided on the vehicle, and the factory radio can be reinstalled when it is time to sell the car.
  

Radio Mounting

Aftermarket radios can mainly be mounted in two ways:

ISO mounting is when the radio can be screwed to existing factory radio brackets, such as in most Japanese cars.

Ring mounting: Most aftermarket radios come with a metal ring that gets mounted to the factory radio hole or aftermarket kit via bendable tabs.  In many cars, dash and trim rings have to be filed to enlarge the radio hole.   Once the ring is installed, the radio slides in and is held by snaps.  In most cases, special tools are required to remove the radio.

Using the Factory Head Unit

Adding amplifiers to factory head units or head units without RCA outputs can be easily achieved with a high-level to low level adapter.  The adapter reduces the level of the signal coming from the head unit's speaker outputs to lower levels that are acceptable for amplifier inputs.  Some amplifiers have this adapter built in for convenience.  The drawback of using speaker outputs is that the signal is not as clear as it would be coming straight from a set of RCA wires.  If the factory unit has distortion on the output, the distortion will be passed along to the amplifier.

Replacing the Factory Head Unit

Many cars with high-end factory systems such as Volkswagen's Atkiv Speakers, GM's Delco-Bose, etc. have amplifiers that require an interface kit to match signal levels, or are best completely rewired.  These kits are usually expensive.   To bypass amplified speakers sometimes existing wiring can be used.  In other cases wires have to be run to each speaker.  Factory amplifiers such as in some Fords use a 5-volt turn on wire instead of the usual 12v.  Even though factory amplifiers can be hooked up to aftermarket radios directly, they can be prone to noise.   Consult a professional before tackling one of these projects.

Getting Better AM/FM Reception

Believe it or not, factory tuners are usually better than aftermarket units.  The most important part of the tuner is definitely the antenna.  If you have a bad or broken antenna, you tuner will not pick up the stations as it should.  If the antenna has to be replaced or upgraded, make sure it is the same length as the original.  The length of the antenna greatly affects reception.  Lower frequencies (AM) are best caught with a long antenna, while higher frequencies (FM) need a shorter antenna.   Car manufacturers compromise a bit, giving you a length that would work best while receiving both AM and FM frequencies.  If you get a short antenna, such as the 1 foot rubber antennas, the FM reception will be poor and AM will be almost non-existent.

Troubleshooting: If you have poor reception, do this simple test:  Try a couple FM stations and a couple AM stations.  If you have no AM at all, but you get FM, then the problem is most likely the antenna.  If your radio can't get neither, then the problem is either a broken or disconnected cable or a bad tuner.  Plug a test antenna to the radio to make sure the problem is not the radio itself.