Enkai's are social functions that usually involve at a minimum some finger food if not a full meal and drinking. They can sometime be followed by a 'Nijikai' or after party.
They can come in two forms - either you are given a set price (this can be as little as ¥3000 or as much as ¥8000 or more), or you're given a rough estimate and you will pay after the event (in this case all costs are normally divided equally).
Additionally they can vary from a set menu with ordered drinks to all you can drink and sometimes even all you can drink and all you can eat.
It's perfectly acceptable to not drink however the cost of the enkai will usually be the same whether you drink or not. If you do drink, be sure to drink responsibly and have a going-home plan! The laws on drinking and driving (or even biking!) are very strict in Japan. If you are caught, you can go to jail for up to two years, be fined, and deported.
What to Expect
When you first arrive at the enkai venue you should greet your coworkers. If there is a coworker/s who are organising the enkai, check in with them and have your name marked off (sometimes this is where you will pay for the enkai).
Some enkai's will be free seating, others will have designated seats and others will be by lottery where you draw your seat.
Once your seated you will normally order a drink. When the drink comes DON'T DRINK IT. Enkai's start with opening words, at the end of which everyone will 'kanpai' or 'cheers' and then you can drink.
At some point in the night people may start moving around and greeting over people - or coming over to see you. You can follow suit.
At the end of the enkai there will be some closing words before wrapping up. At this point you leave, or follow on to the after party (nijikai).
Again, every situation is different, but don't feel pressured to drink - normally there are at least a few teachers who won't be drinking alcohol. Similarly not all teachers will continue on to the after party. Do what you are comfortable with. But no matter what make sure you have a option for going home. Whether that's grabbing a lift from a teacher who didn't drink, another JET, public transport or a taxi. You can not under any circumstances drink and then ride a bicycle or drive a car.