Most GP veterinarians (treating any species, small animals, farm animals, exotic pets, etc) go straight into practice after graduating, without needing to do an internship or residency.
SA rotating internship = this is the "typical" introductory internship. The pay is crap, the hours are long, but many people choose to do this after graduation. 1 year long, you typically "rotate" through different specialized departments (such as surgery, internal medicine, emergency, etc). This is typically required before applying to most residency programs. (with exceptions in things like lab animal - thank Bob - and pathology)
Specialized internship = this is basically what it says, a 1 year internship in a specific specialty (so like a surgery internship), typically happens the year after the rotating internship. This is not necessarily "required" but may end up being practically necessary as people are competing with those who've had that extra training.
Residency = this is the real deal, where you actually complete your specialized training and become eligible for board certification. Most specialty boards require either experience or an approved residency program be completed (or in progress) before you can even sit for the board exam (and it depends on the specialty). This can be anywhere in length from 2 to 5 years (partially depending on the specialty or if additional degrees are combined with it such as a PhD)
All of these things are done after graduation (either right after or, if you decide to come back to it, few years down the road), but none of them are necessary to work with small animals or exotics in a general practice setting. Specializing can eventually increase your salary, but it depends on the specialty. Not to mention the years of additional training at typically much lower pay than you would make if you just joined general practice.
Hope that helps.
