Which leads me to this question: what makes Neil Gaiman so well-known and popular compared to so many other award-winning sci-fi/fantasy writers, and why has his name suddenly come into the mainstream consciousness over the past year?
Like many creators, Gaiman goes through his periods in which he is all-the-rage, and this just happens to be one of them. As others have said, a few recent movies have brought him to the fore as well as the major comic project of Whatever Happened to the Caped Crusader? which appears to wrapping up (yet another) EPIC EVENT IN DC COMICS (*snore*).
Oops, sorry for that editorial...
Gaiman made an intial splash with the Vertigo comic Sandman, a complete reimagining of an old DC character, ultimately related only by name and one small bit in one small story. While a bit uneven in quality over its 75 original issues, Sandman created a set of meta-gods and wove together history, literature and mythology in a set of compelling interlocking stories about Dream, the elemental force of nature that is nightmare and imagination, personified as a skinny Goth guy - and while that sounds lame, it's not. At all. Sandman may be the best high fantasy written since The Lord of the Rings. It also has the charm of being thoroughly post-modern the way that LotR is thoroughly modern, so it kind of captures an era while being emminently readable.
Gaiman has done a variety of multi-media projects, from novels (already mentioned - Good Omens, Stardust, American Gods, Anazi Boys, etc), collections of short stories (Smoke and Mirrors - recommended), children's books (The Day I Swapped my Dad for Two Golfish, Coraline), original movies (MirrorMask - with the great advantage of animation by the fantastic Dave McKean), other comics (including spin offs of Sandman that are charming), tv episodes and adaptations of his work to film.
Gaiman has a very particular style and brand of slightly horrific, slightly hilarious fantasy that makes him a lot of fun. He is very much about character, tends to have pretty rambling plots, and is highly imaginative in terms of his fantasy worlds. None of them have typical swords and sorcery magic, and most involve world mythology in inventive ways. He is the furthest thing from preachy and what themes and allegories may emerge from his work appear to be totally unconscious.
In short, he is simply a damn good storyteller, which is honestly a rare thing these days.
Occasionally, I really love his work.