Jupiter has the most moons by a considerable margin. It's still uncertain how many moons Jupiter has because so many of them are very small, basically strays from the asteroid belt that Jupiter has picked up with its huge gravitational field. Jupiter has four major moons, the "Galilean Moons" which are Callisto, Europa, Ganymede, and Io. In all, it has at least 16 easily viewable moons and many more that have been observed by spacecraft such as Voyager 1 and 2 and Galileo. Some good information on planetary exploration and spacecraft can be found through NASA's Jet Propulsion Laboratory in Pasadena, CA at http://www.jpl.nasa.gov/planets/. Saturn also has a lot of moons in addition to its spectacular rings (although all the big outer planets have rings), including the biggest in the Solar system, Titan. Saturn has at least 17 moons and probably more. Uranus and Neptune also have many moons. Of the smaller planets, Earth has one as does Pluto and Mars has two. Another good place to look for
information about planets and the Solar System is the Planetary Society at http://www.planetary.org/. Where do you think they get the names for planets and moons? Why do you think that it seems like the bigger a planet is, the more moons it has? Have fun exploring the solar system!

The answer to your question will be in any encyclopedia, under "Jupiter" and "Saturn".
Jupiter has 16 known moons; 4 big ones, discovered by Galileo, and 12 much smaller ones. Saturn has 18 known moons, making it the planet with the most known moons.
Both planets also have rings around them, which consist of millions of small rocks and bits of ice. Saturn's rings are much brighter than Jupiters, though. If you use the web, you can find alot of neat information at:
http://seds.lpl.arizona.edu/nineplanets/nineplanets