A less popular name for the common wallet. It basically refers to any thin material that is sewn into a pocket that is long enough to carry any variety of paper money, while at the same time naturally flaps shut. The pocket folds neatly along its midsection, leaving the entire thing roughly square, the size of a pants pocket. They are about the most damn useful thing you can have in your pocket.
Wallets in general, especially billfolds, come with more than just a pocket. You are likely to see in any a small pocket or two for credit cards, a clear plastic pocket for driver's license or other identification. As well as some pocket that will hold small change. Some come with a chain of clear plastic flaps, for pictures. Some come with a chain to prevent theft. (The chain is attached to a buttoned loop of leather which attaches to the beltloop, alternatively the chain can be used for self defense) There is even a billfold designed to hold an iPod Nano.
Generally women do not carry billfolds, they carry a different style of wallet. (Though I make the caveat that this is not an absolute statement) If their purse does not have a pocket for paper money, they may carry a wallet which does not fold at the midsection. The wallet that a traditional, Western woman carries in her purse does not fold at the midsection, but has a flap which folds over and clasps over the pocket of the wallet. This trend is kept alive by fashion, which dictates extremely tight pants for women, thus purses remain useful despite pants proliferation.
Because of this trend, or at least assumed trend, that women do not carry billfolds, men can take care to personalize their wallets without worrying about masculinity. Men make a special effort to turn their wallets into the most useful things they can. Making wallet sized spanners, wallet sized business cards, wallet sized this, that and the other thing. So the absolute variety of things that can be fitted into your wallet has grown and continues to grow.
Personally, I carry my billfold in my front, right pocket, to keep it close at hand and prevent theft. A lot of people keep theirs in their back pocket, my girlfriend does this. Presumably this is to free their front pocket to hold keys or other items and don't want to stab themselves in the butt when sitting down.
I apologize for the overuse of the word pocket in this writeup.