A lot depends on the school and the instructor. Some schools have an organized curriculum and separate classes for beginners. At other schools you'll just learn whatever the instructor feels like showing that day, which may or may not be useful for the total beginner.
In general a typical class will go something like this:
- warm up - this may be some gentle movement to get the blood flowing or it may be a series of intense calisthenics, depending on the instructor
- techniques - the instructor will demonstrate 2-3 techniques, then go around the room to check form and answer questions as you drill the movements with a partner
- rolling - you'll do a number of rounds of free grappling with different partners. This can have different forms - sometimes you might start standing and work for a takedown, sometimes you might start mounted and work to escape, sometimes you might have to defend against punches - but the most common form is grappling only, starting from the knees.*
*(Personally I think that grappling from the knees is not the best training method and I don't use it very often in the classes I teach. However it is the most common type of rolling you see in BJJ classes.)
If you are out of shape, you will end up exhausted very quickly. Even students who are already in good shape typically are exhausted when they first start grappling. Some things that will help - relax, breathe, don't worry about winning or losing, relax some more, breathe, try to remember and attempt the techniques that were shown in class, breathe, relax. Try to make it to the end of the round so you don't cheat your training partner of practice time, but it's okay if you need to roll one round, then sit out one or two rounds while you recover.
After you get home, take a hot bath, do some gentle stretching, take an ibuprofen, and beg your SO for a back rub. This will help reduce the soreness that you will feel the next day.