The ultimate aim of the coagulation factor team is to convert prothrombin into thrombin. This enzyme then activates fibrinogen to form fibrin. Fibrin is an insoluble protein that is polymerised and cross-linked so that it acts as a kind of glue, filling in the defect on the vessel wall (along with incorporated platelets and red blood cells).
The coagulation team consists of about 10 proteins (depending on how you count) that are initially in an inactive form. Each activated factor is then capable of catalysing the next step in the pathway, usually in association with a particular cofactor helper. [Usually, the activated factors are serine proteases, which is just a fancy way of saying that they are a type of protease (enzymes that cleave proteins) whose active site contains the amino acid serine.]
You might be asking: why this elaborate system? Why not simply have one enzyme that is activated and then skips right to the end to generate fibrin? The answer is that each activated coagulation factor is capable of activating several of the next inactivated factors down the line. This means that as you move down the pathway, you get an exponential increase in the number of participants, culminating in much more fibrin generation than if you only used one enzyme.
The coagulation cascade has historically been identified as having two possible starting points, named the 'tissue factor pathway' and the 'contact activation pathway'. It now seems that the latter pathway is very much the poor cousin here, as even a complete deficiency of some of its components fails to lead to a bleeding disorder, although it does prolong the coagulation time as measured in the laboratories.
The tissue factor pathway begins, appropriately enough, when tissue factor - a protein located in the smooth muscle and adventitial layers of the vessel wall - comes into contact with flowing blood. The two will therefore obviously meet whenever the vessel wall is disrupted or injured.
Tissue factor then sets off a series of chain reactions in the manner described above, at last generating a small amount of thrombin. This is sometimes referred to as the initiation phase, and its aim is simply to produce a bit of thrombin. This is because, in addition to generating fibrin, thrombin also activates other coagulation factors (VIII and V). Unlocking these factors leads to an amplification of thrombin-generating potential, called the amplification phase.
As a final touch, factor VIII cross-links the strands of fibrin together (right), adding a robust stability to the entire contraption. (This step explains the rationale behind measuring D-dimers - more on this here if you're interested.)
The final result of the whole process is a thrombus, a sticky plug for the hole in the vessel wall that consists of platelets, a fibrin mesh and trapped red blood cells. There are some wonderful (false-colour) images of thrombi as seen by an electron microscope available, and I highly recommend taking a look at them. One that I particularly like (I'm not sure where it's originally from) is this one, in which you can clearly see all three of the major components of a thrombus (fibrin in blue, red blood cells in red, platelets in purple):
One final note: I have deliberately not gone into detail here, especially about the specifics of the coagulation pathways. It boggles the mind why examiners expect any more knowledge than this at a student or general practitioner level...!