Vermicompost = A mixture of vermicast, pastially decomposed organic waste, and bedding. It may also contain worm cocoons, other insects, and various other organisms.
Vermicast = worm manure (undigested material, soil and bacteria deposited through a worm's anus). Don't get grossed out -- if you dig in the dirt outside, you probably touch this stuff all the time.
Bin composting = dumping material into a structure and either leaving it be (cold composting) or actively aerating, dampening and layering to build internal heat (hot composting).
Tumbler composting = using a barrel mounted on a stand to make turning easier, usually used with quick batch composting, rather than continuous bin composting.
Green waste = material that provides nitrogen, such as grass clippings, shrimp heads, wheat bran, coffee grounds, raw potato skins, apple cores -- in other words, material that tends to be "green, succulent, gooey and dense," to quote Rodale's excellent book on composting.
Brown waste = material that provides carbon, usually stuff that's generally brown or yellow, dry, coarse and bulky, like wood chips, shredded paper, straw and sawdust.