What are the body's two anticoagulant systems?

To ensure that there's no inappropriate clotting or haemorrhage, the body keeps two opposing forces in operation at all times: the coagulation system and a set of anticoagulant systems.  All it then needs to do is to ensure that the right one prevails in the right circumstances.

There are two main anticoagulant systems in place, and the distinction is logical: you can either prevent the clot from forming in the first instance, or else you can break a clot down.  

Doing the former are antithrombin III, protein C and protein S.  Antithrombin III is a serine protease inhibitor, meaning that it foils the plans of enzymes who are trying to cleave other proteins by means of active sites containing serine (serine is an amino acid, remember?).  Recall also that most of the coagulation factors are serine proteases, and you'll immediately see why antithrombin III is a powerful inhibitor of coagulation.  It is made even more powerful (up to 2000 times more powerful, to be precise) if heparin binds to it first - see here.  The main factors inhibited are thrombin, Factor Xa and Factor IXa.

Moving on, protein C is activated by thombin, and simply cleaves factor Va into an inactive form.  It requires protein S as a cofactor.

The other anticoagulant subsystem is the thrombolytic system, which seeks to dissolve clots that have already formed.  The chief actor here is plasmin, which is capable of busting through fibrin (and forming fibrin degradation products, which further inhibit coagulation). It is formed from a precursor called plasminogen.  And what turns plasminogen into plasmin? Well, thrombin for one (it's maturely trying to limit its own activation), and also an endothelial substance called tissue plasminogen activator.  We actually use the latter (as well as related compounds) to break through thrombotic lesions as found in strokes and heart attacks.  The main danger of using them should be obvious: they can cause excessive haemorrhage.