Buff - A solid golden color, no black or white or other in it whatsoever. It is often found in Orpingtons (hence the common term Buff Orpington) but can also be found on Leghorns, Ameraucanas, Cornish, Silkies, Cochins, etc.
Barred - A barring of two colors, one dark and one light, that goes throughout the whole body of a chicken, making it appear checkered or striped. Often people associate this coloring with Barred Rocks, but barring is found on a LOT of different breeds, and even in different colors. A buff chicken can be barred, too. (solid light and dark gold, no other colors)
Bantam is the term for a "small" or "dwarf" chicken. It isn't a breed, and you cannot breed them with "standard" size chickens unless you just want mutts. But, for example, you can have bantam Cochins and standard (or as well cal it, Large Fowl) Cochins.
If you breed a bantam to a normal sized chicken, it's basically a mutt. They come out a midsized chicken, but if you want to show it or something, it cannot be done. Has to be bantam or LF (standard)