I live in the uk
the process is a rather fat looking needle, my cats were done in the back of the neck, but different vets do it in different places.
The price was £14, if you didnt register with the microchip company they charged you if you changed details, if you payed a one off £10 you can change it whenever with some other bonus's which ive forgotten now
Injection is rather simple and not real painful. They just take a needle and insert it under the skin at the neck, in between the shoulder blades. It doesn't take to long, but I'm not sure of cost. Each vet has different methods, costs and microchipping program.
As to process, yeah, what everyone said. Just a big needle, a chip about the size of a rice grain is inserted between the shoulder blades, just like they're getting vaccinated. You can find out the exact info at these sites.
These are the only 2 widely used in the US at the moment, your vet is going to have one or the other. Most shelters and vets have universal readers, but check anyway what's more popular in your aread. Vets' costs vary widely, but the method and program is exactly the same, you're not getting anything special the more expensive the cost.