It actually depends which kind of lessons are available. My baby girl who is now almost 3 years old have been taking "swimming" lessons since she's 3 months old. We bring her to a private pool in the surroundings each Saturday and she gets a private lesson during 15 minutes, without us being in the pool.
In the early stages, the goal is "survival": teach the baby to fall in the water from various heights in various positions (sitting on the poolside, head first, ...) and recover a safe position, on her back, floating for at least 4 minutes. How long this stage is depends on the child, from a few months to a year the teacher said. It took my girl 6 months I think to get her "diploma". Then the process slowly evolves towards what begins to look like swimming movements, but as baby this age don't master all of this nor understand everything they're told to do, this is basically a very repetitive training.
Here are some videos of her during the lessons. The older ones, at the bottom, show some examples of various exercises in the "survival phase". The newer ones, to the top, show some examples of exercices of "real" swimming.
Note that the teacher (and his wife on some videos) has several years of experience with babies, this pool was created by his father who was also teaching kids to swim and is very renowned around here so I didn't look for any other "qualification", which would anyway be useless to you as I live in Belgium :-) I say this because this is definitely not something we could have taught her ourselves, you need a lot of experience to know what to do and when the baby is ready to go further in the "cursus". We are very glad we took her to those lessons, she's really enjoying it, it could be a lifesaver, and it's anyway great to be able to play in the pool with your child on occasions without her crying all the time because her face is wet or so :-)