A single-type advantage (for instance a Water attack against a Ground-type pokemon) will net you double normal damage. The advantages also "add up", so a double-type advantage (for instance a Water attack against a Ground/Rock-type pokemon) will net you quadruple damage. In both these cases you will see the message "It's super effective!" in-game after the attack.
Conversely, a single- and double-type disadvantage will afflict half and a quarter normal damage respectively. Here you will see the message "It's not very effective..." in-game.
Another advantage you can gain is Same Type Attack Bonus (STAB). As the name implies, this increases the power of the move if the attacking pokemon has the same type as the move used (for example a Fire-type pokemon using a Fire-type move). In this case the damage is 1.5 times normal. Again this is added to any other advantages, so a Water-type pokemon using a Water-type move against a Ground/Rock-type pokemon will bag you six times (2×2×1.5) normal damage!