One of the main reasons for using UUIDs is that no centralized authority is required to administer them (beyond the one that allocates IEEE 802.1 node identifiers). As a result, generation on demand can be completely automated, and they can be used for a wide variety of purposes. The UUID generation algorithm described here supports very high allocation rates: 10 million per second per machine if you need it, so that they could even be used as transaction IDs.
UUIDs are fixed-size (128-bits) which is reasonably small relative to other alternatives. This fixed, relatively small size lends itself well to sorting, ordering, and hashing of all sorts, storing in databases, simple allocation, and ease of programming in general.