Off the top of my head:
That I watch/would watch if I had the time? Extras, Hustle, Life on Mars, Spooks, Balderdash & Piffle, Doctor Who, QI, Have I Got News For You.
That I don't watch but other people seem to love? EastEnders, Hollyoaks, the Catherine Tate Show, Little Britain, Casualty/Holby City.
Classic British television: The Office, Only Fools and Horses, Blackadder, Red Dwarf. Um, there's obviously a lot more but I can't remember on the spot.
FiF recommends: Extras - situational comedy based on the work lives of extras with celebrity guest stars (Orlando Bloom and Daniel Radcliffe were particularly hilarious) from the creators of the Office.
The Office - goes without saying; I know they made an American version anyway.
Hustle - crime drama about a group of "long con" con artists.
Spooks - spy drama about the Secret Intelligence Services (i.e. MI5).
Only Fools and Horses - I think it's Britain's favourite comedy, or something ridiculous like that. It's set in the East End and a lot of the humour is based in British culture, though.
Blackadder - starring Rowan Atkinson (Mr Bean), "historical" comedy.
Red Dwarf - long-running sci-fi comedy.
Honourable mention: it's a film, but
Withnail & I is legendary.