The story is, and is not, what someone who hasn't read it thinks it is. Yes, it's about a guy who has a painting that gets old instead of him. But it's much more than that: it's about a soul gradually damning itself.
The scene opens on Lord Henry Wotton (an upper-class cynic) and Basil Hallward (a painter of some note) discussing things in Basil's studio and garden - in particular, a young Dorian Gray, who has been modeling for Basil and with whom Basil is in a kind of love. (There are never really more than the vaguest hints of homosexuality in this book.) Enter Dorian, and Basil completes his masterpiece - a portrait of Dorian Gray.
Lord Henry has a lengthy discussion with Dorian on the wonderfulness of youth, and how Dorian's beauty will open doors for him and then desert him. Dorian, on being gifted Basil's painting, mutters a wish - maybe even a prayer - that he might remain as he is and the picture change instead.
And here's the thing. It isn't so much age, as sin that appears in the picture as the story progresses. It begins when Dorian jilts an actress with whom he has previously been smitten, cruelly, and that night a cruel twist of the lip appears in the picture. Dorian, largely under Lord Henry's influence, gradually goes in for more and more degrading sins - which are not detailed; this is not a pornographic work in any sense - and, despite occasional resolutions to mend his ways, becomes a terrible person indeed: all the while retaining his youthful beauty while the picture grows hideous.
The story's climax is really a few chapters before the end, when Dorian shows Basil what has become of his picture, and its terrible sequel.
Perhaps the finest feature of this short novel is the psychological observation of Dorian's degradation. He blames everyone but himself for it: Lord Henry, a book he reads, the picture itself, even Basil; as well as the victims of some of his sins. There is no mistaking that Dorian is a damned soul, who will take no responsibility for his own damnation. This is, in the end, a supremely _moral_ book.
Of course, everything one knows (or thinks one knows) about Wilde himself colors how one reads his books. One can't help seeing bits of our mythical Wilde in Lord Henry, and in Dorian himself. But Wilde's ability to create and individuate characters overcomes this to a very large degree, and Dorian, Basil, and Lord Henry will remain with me for a very long time.
If you haven't read this...do.