Without much ado about biochemistry (most interesting part for me, but it is another giant talk):
Both clotting pathways work similarly. Each is a row of chemical reactions when one component (they are all proteins which present in plasma in non-activated condition and they all got names which always start from F, for "factor", and continue as a number in Roman numericals, like FII, FX, FXII) activates the next one, which activates the next one, and so forth till FX, or "Stewart-Power factor", becomes active. FX is the central key component which ties extrinsic and intrinsic pathways into one and, with help of other "F"s, makes a plasma chemical named "protrombin", his other name "FII", into its active form "thrombin". Trombin activates another plasma chemical named "fibrinogen" into its active form "fibrin". Fibrin looks like thin and very sticky threads. These fibrin threads stick to exposed tissue, vessel wall and blood cells and form thrombus which cloths the vessel.
Now, several "F" proteins are made in liver. Several of these liver-made proteins, namely FII(protrombin), FIX (which only works in intrinsic part) and that all- important FX guy need vitamin K to be made. Coumadin blocks action of vitamin K, therefore if patient takes coumadin, factors FII, FIX and FX are not made and blood won't make cloths.
Just in case so that all this big system won't start running at a random place and time, organism has at least one chemical blocking each step and every chemical in coagulation cascade. These chemicals do not work like an avalanche and instead stop every cascade step as they go. At norm, coagulation and anticoagulation systems balance each other. If you cut your finger, local activation of coagulation cascade happens and only the wounded vessels are quickly clotted. If some platelets stick to the aterosclerotic plaque in your aorta (all humans have them after early childhood, so do not worry), local anticoagulation kicks in and cloth won't form. But, there are conditions which move balance toward coagulation. These are:
- slow, static blood flow (for whatever reason- immobility, dehydration, low blood pressure)
- inflammation (caused by whatever)
- too much platelets
- various debris swimming in blood (think about metastatic cells, bacteria or microscopic pieces of broken bone)
- and quite a few others. Some people genetically have proteins in their plasma which activate coagulation (if you ever hear about "factor Leiden", that's one of the most common of them, but there are over a hundred of them known for now)
In these cases, coagulation becomes our enemy, because blood starts to get clotted rather randomly and in dangerous places. Since leg veins are wide and "soft", blood flow is normally slower there and cloths happen more frequently. They flow into vena cava, then in right heart, then into pulmonary artery and make PE, which is not good. If a cloth starts to form into arteries which supply heart muscle, there will be myocardial infarction. If it happens in brain artery, there will be an ischemic stroke. In these cases, we can administer drugs named "anticoagulants" (btw, please try not to name them "blood thinners" - they do not "thin" anything, they prevent clotting). Each of them works differently with different "F" factor(s) or with platelets which initiate extrinsic pathway, blocking their work in various ways. They can be "natural" (human blood contains chemicals very much like heparin but there is no way to boost their production in human body so far, so that medical heparin is made from internal organs of domestic cattle) or totally artificial like coumadin.