The biggest source of confusion is that British people tend to understate things, and Americans take that literally.  The classic example is the response to "How are you?" A typical American response is "Great!" or "Good". If someone responded like that to me in the UK I'd have been surprised and considered them arrogant. An appropriate British response is "Oh, surviving". "They haven't killed me yet". When I first moved to the US and responded like that, people would be genuinely concerned. "Hey Jeremy, how's it going?" "Well, it's been worse."  "Oh, are you OK? Anything I can do to help?"

Similarly, one understates how bad things are.  "So, I hear your house burnt down?" "Yes, it's a bit boring." Or "Your wife left you, took the kids and the dog?" "Yes, boring isn't it".

Americans (tend to) respond with something like "It's more than boring, it's terrible!"

It's slightly frowned upon to request a lot of something if you're British. If I'm offered some beer, I might say "Yes, I could drink a little", or, holding out my glass, "You could put a drop in there." Then someone does give me a little, or puts a drop in there.  I didn't want a little, I wanted a lot. But I didn't ask for that, because it's not polite. If a British person asks you for a drop of beer, give them a pint.

Then there are the words. There are lots of words out there (including in other answers to this question). These are some of the less obvious ones.

Pants, in the US, means well, pants.
In the UK, it means underwear. It dawned on me yesterday that "Liar, liar, pants on fire" is used in both places, and therefore has a different meaning.
And sometimes it means no good. 'That's completely pants'.
Ref: pants

The word "merry" is used as a euphemism for being drunk in the UK. This is not the case in the US.
Ref: merry (see 4th definition)

Bollocks means testicles. It is a reasonably offensive word in the UK. Newcastle Brown Ale use it in an ad on the radio in the US. that would not be possible in the UK. "Bollocks" can also be used as an exclamation, and as a description of something that is not very good or not true (bullshit).   Confusingly, if something is "The dog's bollocks", it is excellent - the absolute apex of quality; this is then abbreviated to "the bollocks".  
Therefore:
"Bollocks" = poor, untrue.
"The bollocks" = very, very good.
Ref: Dog's bollocks

Another exclamation is "Gordon Bennett!" (Say it, out loud. go on. Feels good, doesn't it?)
American: "You've dropped sauce on your shirt."
Me: "Oh, Gordon Bennett! Again!"
American: "What? Who's Gordon Bennett?"
Ref: Gordon Bennett!

A garden is a yard in the US (in the UK, it would be a yard if it had no soil or grass). A vegetable garden is a garden (at least I think that's what it is). So one does yard work when one mows the lawn, cuts the hedge, etc. But you you pay someone to mow the lawn, cut the hedge, they're a gardener.

I haven't quite got all the subtleties of stores and shops yet. You go shopping in stores, not (for the most part) in shops in the US.  A supermarket is a grocery store or a market. Maybe it's a supermarket sometimes. A convenience store might also be a market.  An off licence is a liquor store (I think that varies from state to state).  

School often means university / college. So if someone talks about their school in the US, they're talking about the place they went after they were at school. I'm not sure how they distinguish their school from their school. Maybe it's obvious from the context.

Graft is an interesting one - graft, in the UK, is a good thing - it refers to hard work. To describe someone as a grafter means that they work hard, and long hours. In the US it refers to bribery and corruption.

Washing up, in Britain, means to wash dishes. You don't wash up anything except dishes. To an American, one washes up before dinner, meaning to wash your hands. So washing up liquid is the stuff you use for washing up (i.e. washing dishes), but if you ask an American for washing up liquid, they're confused. I still don't know what it's called. I've tried asking for dish washing soap or dish washing detergent, but then I get the stuff you use in the dishwasher, not the stuff you use in the sink.

You cook in the US on a stove, not a cooker. A grill in the US is a large griddle type thing which we'd call something for frying on in the UK. You can also grill on a barbecue. What we call a grill in the US is a broiler, and broiling (grilling) is very rarely done.  I've known people who didn't know how to turn on the grill (broiler) on their cooker (stove). A frying pan is a skillet.

The worst place for not being understood is Home Depot. Lots of things have different names. Everyone knows that a spanner is a wrench. But putty (or what I think of as putty) which is stuff for holding glass in windows is not putty, it's got another name (which I've forgotten), and what Americans think of as putty is something different. What is Plumber's Putty?Why do plumbers need to hold glass in windows?  I don't know.

Masking tape is painter's tape. Plasterboard is drywall. White spirit is mineral spirits. I don't know what woodstain is (the kind made by Ronseal, which does exactly what it says on the can), it's not what's sold as wood stain.  Breeze blocks are cinder blocks. Silicone sealant is caulk.

On one occasion I wanted a threaded rod from Home Depot. There's no way I was going through the procedure of finding someone to ask, and then trying to explain what I wanted (I had enough trouble when I wanted a rubbish bin - eventually I gave up and just looked). But after wandering the aisles of Home Depot for quite some time, it turned out that the American for threaded rod is threaded rod .

A rubber is an eraser. Children regularly take rubbers to school. (There was considerable debate in our family when I was a child about whether this was because you rubbed with them, or they were made of rubber. The conclusion was that tyres were made of eraser.)