A film called Carnage that came out last year is outstanding for this. The whole film revolves around the conversations of two couples discussing their children after one of them hit the other with a stick. It sounds boring, but it's outstanding and the very funny.
http://www.imdb.com/title/tt1692486/?ref_=fn_al_tt_1
Another similar film is Abigail's Party. This is older (released 1977) and British (I don't want to make the annoying assumption that American's "won't get it", I just thought I'd mention as it may affect your perception of it) but also very funny, and similar to Carnage.
http://www.imdb.com/title/tt0075622/?ref_=fn_al_tt_2
These are two fairly similar films that revolve entirely around the dialogue of the characters. Both are worth checking out, but Carnage is probably the more enjoyable for a modern audience. Funnily enough, both were originally written as plays.
For something with a bit more action but also with great dialogue, I'm going to mimic others and say basically anything written or directed by Tarantino. Jackie Brown and Reservoir Dogs are particularly dialogue heavy.
Edit: Forgot to say the Coen brothers are also especially good at writing dialogue. The Big Lebowski is my favourite (seriously, watch it) but Fargo also deserves a mention for this.
Edit 2: This will be the last one, I promise haha. I also want to add In Bruges, which is another heavily dialogue driven film. Don't watch the trailers, it was marketed completely wrong. It's a good, quite dark film with touches of black humour, as a hitman tries to come to terms with the accidental murder of a child, whilst in Bruges with his partner.