The Dempster Dumpster had a standard form factor, so that the forks of a Dempster Dumpmaster truck could pick it up and dump it into the back.[1] The patents have expired, so now anyone can make compatible rubbish bins and trucks. The standard form factor lives on for compatibility with the installed base. Properly, the term "Dumpster" refers only to containers compatible with that system.

There's also the Dempster Dinosaur, for when a Dumpster just isn't big enough.[2] That form factor is called a "roll off".

The Dempster brothers solved the problem of collecting, lifting, packing, and moving a lot of trash without hand labor, and dominated the industry for decades. Dempster, the company, seems not to have survived. The "Dumpster" trademark was allowed to lapse. From the USPTO:

    DUMPSTER (EXPIRED) 
    CONTAINERS FOR RECEIVING, TRANSPORTING, AND DUMPING MATERIALS OF VARIOUS
    KINDS-NAMELY, REFUSE, TRASH, GARBAGE, SCRAP, DIRT, AND ROCKS.
    FIRST USE: 1936-09-01. 

The generic term is "rubbish bin".

[1] http://www.classicrefusetrucks.com/albums/DE/DE07.html [2] http://www.classicrefusetrucks.com/albums/DE/DE05.html

A skip? A bin?

I mean I know that doesn't replace that name to your ears now, but if that word was never made up others would have been used instead, as they do in some other anglophone corners of the world.