We bought this piece of furniture (from a charity shop) for our kitchen last week, and I am learning to call it a Welsh dresser. It is a low cabinet with an open case of shelves on top. It has a little surface in front of the shelves where one could, say, slice bread. I was stymied in trying to find an American equivalent for this--which may go to show that I am suffering attrition of my native dialect. I've been calling it a cabinet when describing it to Americans, but the more specific name for it is hutch (as I found after searching US furniture retailers' sites). This is a word I know, but perhaps I never got to know it well enough, as I'd never lived in a house with one before now. That's my excuse for forgetting it, at least. Better Half protests "Rabbits live in hutches!" Strictly speaking, according to the furniture sellers, it's the top part that's the hutch, but since I didn't know until recently that the top and bottom halves were separable, I've always assumed that the whole thing is a hutch. (Without the top hutch part, it would be a sideboard--as long as it's in the dining room or kitchen.)
BH often reduces Welsh dresser to dresser, as in We bought a dresser for the end of the kitchen. This, to me, is weird (technical linguistic term), since I think of a dresser as belonging in a bedroom or dressing room. I suppose one could dress a chicken on a Welsh dresser...
While we're in the kitchen...the built-in work surface is called the counter or countertop in AmE, but tends not to be called this in BrE. Location-wise, it's generally referred to as the side, as in The plates are on the side or Cut the carrots on the side (which, of course, is ambiguous). You wouldn't, however, say I bought a new side for the kitchen or The side is formica. Referring to the thing, instead of the location, it can be called a worktop or work surface.
The things with doors above and below the counter/worktop are what I would call kitchen cabinets. Better Half calls them kitchen cupboards, which also works in American English. In BrE, it seems, a cabinet is free-standing.