Is it likely that a given car stereo will work in a given car? Are there a number of different form factors (sizes)? Are some more common than others? Are there a variety of connector schemes?
I'm thinking of the stereo itself, rather than the speakers. Presumably I would want to make sure the speakers are not going to be blown by a much more powerful stereo. Are there other compatibility issues between stereos and speakers?
What other factors need to be taken into consideration?
Many sites including Crutchfield have charts that will tell you what equipment fits in your car. You need to make sure that the radio will fit where the factory radio comes out, there are adapters that make installation very easy. You will need the correct electrical adapter so you can plug the aftermarket stereo into your existing wiring harness. You should never cut off the old plug and spice in the wires.
Adapter kit's like the one below allow you to securly mount the radio in the vehicle using the factory location and screw holes.
Wiring adapters like the one below let you connect the aftermarket radio wires to the adapter and plug the adapter into the factory harness.
Using both of the adapters above allow you to install the radio without making modifications to the car. In the event you sell the car you can simply install the factory radio again.
Most aftermarket stereos should conform to the DIN standard (ISO 7736), which defines the height and width of the enclosure. As Larry says, you can then get adapters to fit these into those cars that don't conform to the same standard.
I don't know if it's the same in the states, but most European cars I have come across have a DIN socket, but often concealed behind a manufacturer-specific front panel - You can then simply buy a new front panel with a DIN-sized hole in it.
The wiring is again standardised (ISO 10487), but a lot of manufacturers, partiularly on older cars, use their own connectors - You can normally buy a double ended adapter cable to go from the ISO connector on the back of the stereo to the OEM one in the car.
Unfortunately the DIN standard is less and less supported, so it is becoming much harder - you may need to make your own mounting brackets.
In terms of wiring, there are many standards for wiring pin-outs, however the vast majority require just a continuous and a switched 12v in and GND so you can either use an aftermarket adapter kit (mentioned by @Larry) or wire one up yourself.
As the other posters already mentioned, for most popular cars you should be able to get both adapter/mounting plates so you can mount a DIN-sized stereo and the necessary adapters to connect the headunit to the original wiring loom. I would strongly recommend getting the latter rather than butchering the wiring loom as it makes installation and fault finding easier than the home grown 'just crimp them together somehow' approach.
In general the speakers should be compatible with the headunit, with one exception - the Bose systems that you find in some cars have a separate amp per speaker and use a different level output at the headunit (and different impedance on the speakers, IIRC they're around 1Ohm). For some cars you can get adapters that allow you to use a 'regular' headunit with a Bose system, for other cars you'll have to pull out the whole thing and replace headunit, speakers and wiring.
