It's a common thing to do anymore. With the accessibility of used motors from their domestic markets, dropping a more powerful motor into your car is becoming increasingly cheaper. This in no way means a motor swap is easy, economical, or even a good idea.

Modifying a car to the point of doing a motor swap requires two things: a huge amount of time, and an even larger amount of money. This is the basis of everything.

1. "Will X motor fit in Y car?
Yes. Theoretically, with enough time and money, anything is possible. The real question is, is it feasible? A LS2 powered Mini Cooper sounds like a great idea. But the fact is, it's not really a good idea. Most people don't go to this extreme. More commonly, it's a motor from a different car by the same manufacturer- IE, a B16 swap into a Del Sol or a 1.8T into a Mk2 GTI. These are relatively easy, but there is still a lot involved.

2. "What all will I need to do it?"
This is totally dependent on what you're doing. If you're doing a swap that involves putting a motor that was available in the same chassis of car (IE, a VR6 swap into a Mk3 Jetta/GTI), then it's not a whole lot. At the bare minimum, you'll need the motor, transmission (if you plan on using that), axles, driveshaft(s), transfer case, and differentials. For the electronic side of things, you'll need the PCM, the complete wiring harness from the donor car, applicable control modules from the donor, and possibly the instrument cluster.
Then there's the other "little" things like motor and trans mounts, shift linkages, brake components, spindles, etc.
If you plan on inter-brand swapping, you'll need custom fabricated engine/trans mounts, custom drivetrain components, and an incredible amount of knowledge of electronics. And having a shop manual for both your car and the donor car (and any 3rd or 4th cars that you're using for components) will save you hours and hours of headaches.

3. "Wait, you said electronics. What exactly is that going to involve?"
Assuming your new motor isn't carbureted, you'll have fuel injection. This is where the donor car's PCM comes in. (The PCM comes in a lot, but this is its most important function- it makes the motor work.) You'll need to remove all of the engine wiring from the old motor, and replace it with the new harness from the donor motor. This may or may not include splicing wires. Once you get that out of the way, if you want to drive this car on the street, you'll need to wire up a speedometer, and you'll probably want a tach. Methods of getting this information (especially on older cars) can vary greatly. It will depend specifically on your swap. Most modern OBD-2 cars use a Cam or Crank Position Sensor to signal the engine's RPM, while most use a gear on the transmission for the speedo read-out.
But it may be different. And you may have to splice and solder wires to make it work with your older dashboard.
If you plan on keeping the car's original PCM to run the accessories (mainly the interior), then you'll need a stand-alone setup to manage the engine. Also, if you plan on having 2 PCM's in the car, you'll need a stand-alone program to translate different manufacturer's code so they can communicate. I've seen it done this way once or twice, but I don't believe it's the best way to do it. It's a lot of extra wiring that I don't really see as necessary.
Just remember. Installing a motor isn't anything like installing a head unit in your buddy's Cavalier. Take your time, do it right. If you mess up on the wiring any number of things can happen. The best case scenario is you get it done and it won't start. Worst case is you burn your house and car to the ground.
Oh yea, and you'll probably be re-wiring the entire fuse box for the whole car.

4. "Yea, I got all that down. What am I going to need tool-wise?"
How's about $10,000 worth of stuff? Just to get started, anyway. Ratchets. Sockets. Screwdrivers. Hammers. Wrenches. Hydraulic jack and stands. Engine stand. Engine hoist. Air gun and compressor. A good shop light. Electronics require the aforementioned shop manuals, but you'll also need a good soldering iron. Don't cheap out on this part. That $10 will work, but that $25 or $30 one will work better. A good voltmeter. High impedence is a must. You don't want to cook a $2500 JDM PCM, do you? Then there's the almighty manufacturer specific tools. Most of these can be ghetto-rigged by using a combination of other tools, or you can make them yourself if you're so inclined.

5. "Cool, how long should it take?"
This is completely dependent on you and your abilities. I know of people who can do a whole MK3 VW VR6 swap in under 9 hours. I also know of people who got in over their heads and now their cars are members of "Club Never Run Again." If you have any doubts about your abilities to wrench on your car, then you probably should put that 10mm wrench down and consider paying someone more competent to do the swap for you.
Also, length of time depends on the difficulty of the swap. If you have to have custom parts made, it's obviously going to take as long as your machine shop takes to get parts made for you. If you are having parts made to exact specs (IE, a driveshaft for a Miata/LS1 swap), measure more than once. Accuracy will save you time in the long run. And it will save you money.

6. "I still wanna do this! But how much is it going to cost?"
A lot. Definitely more than you'll budget. Probably a lot more. Take whatever you think it "should" cost. Then double that. Most swaps are using motors pulled from wrecked donors or donors that are being parted out for one reason or another. This means the engine already has miles on it. At the minimum, I'd recommend pulling the plugs, and checking the piston tops and cylinder walls with a borescope (look! another tool to buy!!!). When you drain the old oil, take a sample of it to somewhere that can analyze it for you. Check the compression on the motor. Take the oil pan off and check the rods, bottom of the pistons, crank, and main caps.

7. "What if I want to do a turbo/supercharger nitrous set up, too?"
Spool down there, turbo. While you could go ahead and do the forced induction install at the same time, I'd recommend against it. If you do both at the same time, what happens if (when) the car doesn't start or run right? Did you mess up the turbo install? Or is it something base engine? You don't know because you tried to do too much at once. Get the car running first. Then worry about making stupid amounts of power with it.

8. "Sounds good! Any other advice?"
-You may want to take the whole engine apart, including the head, to make sure nothing is bent. Just because the person you bought the engine from said it turned over before he took it out, doesn't mean the engine turns over. Verifying that the components are in good condition before installing them will save you time and money in the long run.
-Replace all the seals in the engine, especially if it's higher mileage. A few dollars now is worth a lot if it prevents failures 6 months or a year down the road.
-Plan, plan, plan. Take your time.
-Document everything. Take pictures of how things look in the donor car if possible. It will give you a point of reference later on.
-Have fun. Cars are a hobby, but they can be frustrating if you're working alone. Invite some friends over to help, order some pizzas, drink some beers.
-Find a forum, or forums, specific to the car(s) you'll be working with. When it comes to specific questions and problems you'll have, members there have infinitely more amounts of wisdom than the kid who works the counter at PepBoys, or even on a generic automotive forum.
-Details, details, details. The more time you spend on the swap, doing things the RIGHT way (which is rarely the fast way) will result in a cleaner install, with less reliability problems. It is absolutely inherent that a swapped car will be at least slightly less reliable than it was when stock. You can reduce this by taking your time, especially when doing the wiring. Don't just splice connectors, tape them up, and shove them in a corner. Wires don't like corners, they hate electrical tape, and will eventually corrode and fail. Then you'll have real nightmares.
-Label things. Especially in your now-hybrid fuse box. Make your own wiring diagrams if you need to. Try to color code wires you have to run on your own to what was there stock. It will make it easier to use your shop manuals later on.


I'm not saying motor swaps are impossible for all but the wealthiest and most knowledgable. But know what you're getting in to before you're neck deep in it and have no idea what you're doing. You'll be stuck with a car that has 2 engines; neither of which work.