A covenant is a way of establishing a relationship. In a covenant you make promises to one another and pledge to live a certain way.
For instance, when a couple loves each other they will enter into the covenant of marriage. They promise to love one another (and only them!) all the rest of their lives.
The Bible teaches us that God relates to man by means of covenants. In the very beginning of history God covenanted with Adam. Essentially God said, "Adam, I promise to let you live as long as you obey me." We call this the "covenant of works" because the life Adam had was based on his obedience.
Unfortunately, Adam didn't uphold his end of the covenant. As a result, he deserved to die. But in Genesis 3 we see that God doesn't exact that punishment right then and there. Instead of killing Adam on the spot God mercifully instituted a new covenant. This is what we call the covenant of grace. Now Adam could live and expect salvation, not based on what he would do, but on the basis of the works of Jesus Christ.
The same holds true for all men today. All men are liable to death because they are covenant breakers and are unable to live up to the requirements God lays out in his law. Yet, in the covenant of grace the Lord freely offers unto sinners life and salvation by Jesus Christ. All he requires is that we have faith in Christ and believe that he will indeed fulfill his promise.