What is the American term for the British word 'skip' in terms of the yellow thing that you throw your rubbish into and it gets picked up later and taken away?

Sounds like you're talking about a debris box, which is used for community garbage collection, and general construction applications. They're also referred to as skip boxes in some states.

The term Dumpster was originally patented by a fellow named Dempster in 1935, ('Dempster Dumpster') and has since been used to describe any similar carry-off debris container.

Skips are large, open topped, usually upside-down-trapezium-shaped metal crates. They are about 6 ft tall and 10 ft long. Nothing like a dumpster at all.
http://www.lsswaste.co.uk/

http://www.selectaskip.co.uk/

If you wait for the first one to load, there is a little animation of a skip falling from the sky. And then an information page.