I've found JabRef, http://jabref.sourceforge.net/ , to be the best tool (for me) for organising a BibTex database, and linking / organising all of my downloaded pdfs.
The biggest advantages of JabRef are that it is genuinely cross-platform, being written purely in Java, and that the native format for the database is BibTex, which means that if I decide to change my methodology then I can straightforwardly use my database with other programs. Storing the database as a BibTex file also means that I can directly edit the database, e.g. doing search and replace with an editor.
It has all of the functionality I need, and in particular it is possible to apply labels to each paper for categorisation, either manually or via keyword searches.