Question: How many moons are there of Saturn and what are their names?  

Answer: As far as we are aware, Saturn has 22 known moons, also referred to as
satellites (more about this later).

We know the names of 18 of them.
In order of radius size (largest first) they are:

Now for the other four!

An announcement was made on 26th October 2000 at the Division for Planetary Sciences, American Astronomical Society, in Pasadena, California that four new outer moons of Saturn had been discovered. They are irregular satellites and have long and looping orbits, unlike the orbit of an inner moon which is almost circular. They were discovered by Gladman, Burns, Nicholson, Petit, Scholl, Kavelaars, Holman and Marsden. They think they may have discovered still more moons of Saturn.

We do not know the names of these satellites and it appears the only thing known about them is how bright they are.