CiteULike is quite nice for this.

Connotea is a similar "personal research library" service but it doesn't let you store PDFs, just links to articles.

Zotero (a Firefox extension) is fantastic for this.

Mendeley is good for this, and specifically designed for managing a library of academic papers. It supports tagging and full text searches, as well as some half-baked "social" features which can be safely ignored. The most useful feature for me is that it can watch a directory for new papers, and add them to its library as well as my directory tree (author/year/paper). It can also maintain a bibtex file for the entire library which is handy for citations.

Depending on how many you save per month, a premium Evernote might be useful for this - it allows you to organize and search inside PDFs, among other things. A premium account has a limit of 1 Gb of saved stuff a month, but that seems pretty reasonable to me.

(Edit to clarify: There is no limit to the amount of stuff you can save in an evernote account, just a limit to how much you can add to it per month. There does not, unfortunately, seem to be a way to add a backlog of things all at once, though, so it might not be the best option for people with extensive collections already.)

Poorly. I have a NAS that holds tens of thousands of PDFs. They are organized in folders, one for each letter of the alphabet, by last name of the first author. It begins:

AAAI - Interim report August 2009.pdf
Abbey - Charles Taylor.pdf
Abbott - A note on the nature of water.pdf
Abbott - Fodor and Lepore on Meaning Similarity and Compositionality.pdf
Abbott - Realism, model theory, and linguistic semantics.pdf
Abbott - Water = H2O.pdf
Abbruzzese - On using the multiverse to avoid the paradoxes of time travel.pdf
Abdel-Khalek - Happiness, health, and religiosity, significant relations.pdf