UK US
cupboard closet (built into the wall, in the UK often under the stairs*)
cabinet cupboard (in the kitchen, fixed to the wall)
dresser cabinet (in the dining/living room, used for display)
chest of drawers dresser (in the bedroom, for clothes)
wardrobe wardrobe (yay!)(takes you to Narnia)
Not all of those are strictly UK vs. US though.
I'm a born and raised Brit and my definitions are:
- Cupboard: what you find in the kitchen (or under the stairs (under-stairs cupboard), or in a bedroom (i.e. a toy or games cupboard), or on the landing (airing cupboard)
- Cabinet: usually found in the bathroom where you store medicines etc., often with a mirror on the front. Or it might be in the living room, storing ornaments or other knick-knacks (also known as a dresser).
- Dresser: chest of drawers in the bedroom, often with a mirror on top (or a lower version like a desk that you can sit at).
- Chest of drawers: in the bedroom, where you store clothes (also called a dresser)
- Closet: in the bedroom, built into the wall (often only called a closet if it stores clothes, otherwise it's called a cupboard - although some people call them cupboards instead of closets anyway)