Anselm is probably best known for his "ontological argument" for the existence of God. He believed that faith is necessary for true understanding ("I believe in order to understand") and that God's existence is self-evident.
In theology, though, he is best known for a short work called Cur Deus Homo (lit. "Why the God-Man") which is about why God had become incarnate as man in the person of Jesus Christ. He was one of the first to put the theological pieces together to realize that what Christ did on the cross was to pay the penalty due to God for sin. Previously, there was this vague idea that Christ's death was for our salvation and that we are saved by faith in Christ alone. What Anselm did was to fit these together in the context of Christology to show that Christ is our substitute before God.