What are the requirements to convert an FAA certificate to EASA/JAA? If a pilot holding a US certificate wanted to convert it to an equivalent EASA (JAA?) certificate, what training, flight experience, and exams are necessary to obtain equivalency?

Does the process differ for private, instrument, commercial, ATP, and instructor ratings?
As @Pondlife states, this will depend on each member state's implementation of the EASA rules but I would expect that the implementations would be similar -- otherwise what is the point of EASA?  (That would be a good question except it would no doubt be closed as being too "opinion based".)

I've added a link to the relevant UK documentation and marked this as a Community Wiki answer so that hopefully others can expand on this for other states and perhaps add personal experiences.

UK

CAP 804 Section 4, Part Q deals with "Validation and Conversion of Non-EASA licences issued by States other than the UK".