Escapepod - ~weekly podcast of short science fiction stories. Archive of all podcasts to date, back to 2005. Very Excellent site, with good stories and very good narration of these stories.
Pseudopod - 'sister' podcast to Escapepod, horror stories here instead of sci-fi.
Podcastle - another 'sister' to Escapepod. This podcast does short fantasy stories.
Well Told Tales - This podcast does short scifi/fantasy/horror stories. Also has an archive of podcasts.
Starship Sofa - A podcast from the other side of the Pond. Excellent readings from some well know authors.
Drabblecast
Novels, Novelettes, Short Stories
Librivox - This website has volunteers that read stories that are in the public domain (out of copyright). There is a very wide selection of genres, including scifi.
Podiobooks - Excellent website with a wide selection of genres.