ChrL nailed it. The only bit to add is that on BA upgrades and awards come out of the same bucket (it's implied in the above post but is worth expressly stating). Thus, if an award is available on BA's site, then an eligible ticket can be upgraded with AAdvantage.

PS: The above post also notes that you should try to book into J... That's true if you want to upgrade to Club World, but the key here is to search for whatever cabin you're interested in upgrading into. If you're trying to go from World Traveller to World Traveller Plus, you'll want to search for World Traveller Plus inventory... And so on.