There are 15: Fire, Flying, Water, Psychic, Ice, Dragon, Grass, Poison, Bug, Dark, Steel, Electric, Ground, Rock, and Normal. Each of these is strong against some kinds and weak against others' attacks. For example, Fire-types are strong against Ice-, Grass-, Bug-, and Steel-types, but weak against Water, Ground, Rock, and Dark-types.
Also, Pokemon of a certain type have similar stats (e.g., Electric-type Pokemon are fast, Steel-types have high defense).
Here is a chart representing some type's strengths and weaknesses:
Flying: Strong against Fighting, Weak against Electric
Water: Strong against Fire, Fighting, Rock, and Steel, Weak against Electric
Psychic: Strong against Fighting, Poison, Ground, and Ghost, Weak against Psychic
Ghost: Strong against Normal and Fighting, Weak against Dark and Psychic
Fighting: Strong against Rock, Ground, and Normal, Weak against Water, Psychic, Flying, and Ghost
Electric: Strong against Water and Flying, Weak against Rock, Ground, and Grass
Grass: Strong against Water, Rock, Ground, and Electric, Weak against Fire and Poison
Fire: Strong against Steel, Grass, and Ice, Weak against Water, Ground, and Rock
Rock: Strong against Fire and Electric, Weak against Water, Grass, and Fighting