In prayers of adoration or worship, we exalt the greatness of God, and we acknowledge our dependence on Him in all things.
Through prayers of petition we ask God for things we need—primarily spiritual needs, but physical ones as well. Our prayers of petition should always include a statement of our willingness to accept God's Will, whether He directly answers our prayer or not.
Prayers of intercession – praying for others – is another form of petition, but they are important enough to be considered their own type of prayer. As the Catechism of the Catholic Church notes (# 2634), "Intercession is a prayer of petition which leads us to pray as Jesus did." In a prayer of intercession, we're not concerned with our needs but with the needs of others.
Perhaps the most neglected type of prayer is a prayer of thanksgiving. While praying “Grace” before meals is a good example of a prayer of thanksgiving, we should get into the habit of thanking God throughout the day for all the good things that happen to us and others. Consider including the Grace After Meals prayer as an excellent practice.
Prayers of praise acknowledge God for what He is. As the Catechism of the Catholic Church (# 2639) notes, praise "lauds God for his own sake and gives him glory, quite beyond what he does, but simply because HE IS. It shares in the blessed happiness of the pure of heart who love God in faith before seeing him in glory." The Psalms are perhaps the best-known example of prayers of praise.