The planet with the most moons in our Solar System is Jupiter; as of 2009 the planet has 63 confirmed moons. However it is important to note that this number may change in the future as more moons are discovered.
Saturn has 61 confirmed moons so if more moons are found on Saturn it may become the planet with the most moons.
Jupiter has 8 regular moons: 4 big spherical moons and 4 other smaller moons that orbit closer to the planet. The other 55 moons are irregular satellites that are also quite small.
Source: http://www.universetoday.com/38187/which-planet-has-the-most-moons/