For non-sorted the best choice, more than nine times out of ten, will be: ArrayList, HashMap, HashSet.
Vector and Hashtable are synchronised and therefore might be a bit slower. It's rare that you would want synchronised implementations, and when you do their interfaces are not sufficiently rich for thier synchronisation to be useful. In the case of Map, ConcurrentMap adds extra operations to make the interface useful. ConcurrentHashMap is a good implementation of ConcurrentMap.
LinkedList is almost never a good idea. Even if you are doing a lot of insertions and removal, if you are using an index to indicate position then that requires iterating through the list to find the correct node. ArrayList is almost always faster.
For Map and Set, the hash variants will be faster than tree/sorted. Hash algortihms tend to have O(1) performance, whereas trees will be O(log n).