How does Christianity work with all the biblical covenants? One major theme of the Bible is the number of covenants between God and humanity. How do Christians manage to work them out? Are the covenants prioritized from highest priority to lowest priority, or are they all of equal weight or importance? For Christians who may hold a dual-covenant theology, how does that work out? Do they really submit themselves to the Torah?
Answering from the Reformed perspective:

There are essentially two covenants. The first covenant was with Adam, and is called the covenant of works. Adam was bound to obey the command not to eat of the fruit of the Tree of the Knowledge of Good and Evil. If he obeyed, he would live. If he disobeyed, he would die. Adam disobeyed, and his sin, because of Adam's representative nature (see Romans 5), is passed on to all people by ordinary generation (that is, everyone but Jesus Christ). 

The second covenant is the covenant of grace. Most of the Bible is spent in tracing out this second covenant. Most of the OT covenants (except that of works) are in this line. They just keep getting bigger and bigger in scope. So you have:


Adamic Covenant (second one): Individual
Noahic Covenant: Individual + family
Abrahamic Covenant: Individual + family + descendants
Mosaic Covenant: Nation
Davidic Covenant: Kingdom


Finally, you get to the New Covenant in the grace line, which is so powerful it's retroactive. 

So what to do with all these covenants? Well, first of all, I would point out that if anyone could perfectly keep the terms of the covenant of works, that person would be saved. Big if. The fact is, Adam's sinful nature is passed on to us, which rules this possibility out from the get-go. How many sins does a person have to commit in order to go to hell? Zero, because of the sinful nature. 

So if we can't get to heaven with the covenant of works, that leaves the covenant of grace. Now, the New Covenant that Jesus instituted rather supercedes the previous ones, and indeed is the covenant to which all the other OT covenants in the grace line were pointing. It fulfills them. Therefore, because Christ is the perfect sacrifice, it would actually detract from His sacrifice to continue offering up the OT sacrifices. 

The definitive work on the subject, so I'm told (I haven't read it, but I've heard it's good), is The Christ of the Covenants, by O. Palmer Robertson.