Only use Vector when you have no choice, when interacting with code that requires it.
Vector was created before the Collections Framework was added in 1.2, and was then retrofitted into it. ArrayList was created as part of the Collections framework from its beginning. Though both classes implement List, Vector also has a bunch of methods that are not part of the standard List interface, going back to its days before the Framework was defined. Vector will function just as well as ArrayList in most or all situations, but the general consensus now is to avoid it. If somebody sees it in your code, they're going to wonder why you're using it--whether you had a good reason to use it, or if you just didn't know that ArrayList is now preferred.
Use ArrayLists when you need any old ordered collection, or when you need good random-access times.
Use Vectors when the API you're working with specifically needs them. I believe this is still the case for some classes in the Swing API, for example. *Never* use them when you're working with new code.
Other alternatives are always better than Vector. For regular use, ArrayList is better. For synchronized use with many edits, use Collections.synchronizedList(arrayList). For synchronized use with many traversals, use CopyOnWriteArrayList.
The mind is a strange and wonderful thing. I'm not sure that it will ever be able to figure itself out, everything else, maybe. From the atom to the universe, everything, except itself.