There are five styles of kickboxing: Semi-Contact, Light Contact, Full Contact, Low Kicks and Musical Forms.
Semi-Contact uses light strength punches and kicks to score points by making contact with the head or body above the belt (waist). The participants must not hurt each other. A match consists of 1 or more rounds of sparring. Sparring begins and after a point is scored it stops. The point is counted and then sparring begins again.
Light Contact is similar to Semi-Contact except that sparring is continuous and judges count points scored as the round goes on. These points are counted at the end of each round.
The Full Contact style basically differs from Light Contact in that you use full power in your kicks and punches. You can also aim to knock out your opponent. Additionally to this, in the Low Kicks style you can kick to the thigh (hence the name).
Musical Forms is a very different style of kickboxing. A routine of techniques and moves, possibly including specialist weapons, is performed to music by a solo martial artist.