OK. Saved the best for last. The Thick of It is the best British Comedy in the history of ever.
Warning: VERY strong language.
The Thick of It is spectacular. I'm happy to see someone else posted it here already. There's no character on any show I can think of quite like Malcolm Tucker. (also, Salome, are you British? The Thick of It fans are few and far between in America. I bought the series on amazon.uk, since it isn't available in the States, and watch it on my region-free DVD player. I think it's getting a release here soon.)
Robin: Malcolm, do you know... how to fix a paper jam?
Malcolm: I don't know, kill a kid an hour until it sorts itself out?
Apparently Peter Capaldi and Craig Ferguson used to be in a punk band together. I saw him on Ferguson, and he told a story about a night he and Craig met up after both taking LSD, and neither one knew the other one was on acid.
Darkplace is exceptionally funny as well. The Mitchell and Webb Look and Peep Show are both pretty funny and on Netflix instant.
I was discussing British comedy with another member on vent and we came to notice (I believe I did with my superior sensing skills) that British comedy is based mostly on metaphors. They make metaphorical jokes.