An enkai is basically a Japanese drinking party, just slightly better organized than that definition would imply. Enkais usually revolve around some sort of business, community organization or group. They basically boil down to an office party which is held for certain events (and not in the office).
For example, if a new employee joins a company, his department will throw an enkai for him at some point during that first week. The same goes for when someone is about to leave a company (especially after a long period of service). Its just a big congratulatory hug and bonding experience for all members of the group, something which can be very important in Japanese culture.
Enkais are usually held at restaurants or izakayas, and many cater specifically to these events. The food provided obviously depends on the quality of the establishment, but it is usually some pretty goddamn oishii stuff. The constants at these shindigs are bottomless supplies of beer, sake and edamame. Payment is usually handled in full and up front before the proceedings.
Following the enkai there is usually a "second enkai" (nijikai) and even a "third enkai" (sanjikai) but it is really just group bar hopping. As such, it is a pretty rare group if you manage to make it to a yonjikai or beyond. Have fun stumbling home, and lord help you if you have to ride a bike.
Update: gn0sis has informed me that nomikai is almost an exact translation of "drinking party."