What is Coagulation? How does our blood clot?

Coagulation is a complex process by which the blood forms clots to block and then heal a lesion/wound/cut and stop the bleeding. It is a crucial part of hemostasis - stopping blood loss from damaged blood vessels. In hemostasis a damaged blood vessel wall is plugged by a platelet and a fibrin-containing clot to stop the bleeding, so that the damage can be repaired.
Coagulation involves a cellular (platelet) and protein (coagulation factor) component.
When the lining of a blood vessel (endothelium) is damaged, platelets immediately form a plug at the site of the injury, while at the same time proteins in the blood plasma respond in a complex chemical reaction, rather like a waterfall, to form fibrin strands which reinforce the platelet plug.
Primary hemostasis - when the platelets gather at the site of an injury to plug (block) it.
Secondary hemostasis - proteins (coagulation factors) act in a series of chemical reactions to strengthen the plug and allow healing to begin.

What is a platelet?

A platelet is a disc-shaped element in the blood that is involved in blood clotting. They aggregate (clump together) during normal blood clotting. They are classed as blood cells, but are in fact fragments of large bone marrow cells called megakaryocytes.

What is fibrin?

Fibrin is an insoluble protein involved in blood clotting. Fibrin is deposited around the wound in a form of mesh to strengthen the platelet plug. The whole thing dries and hardens (coagulates) so that the bleeding stops and the wound then heals. Fibrin is developed in the blood from a soluble protein, fibrinogen.
When platelets come into contact with damaged tissue thrombin is formed as a result of a series of chemical processes (coagulation cascade) that culminate in the formation of fibrin from fibrinogen.

Coagulation factors (clotting factors)

Coagulation factors are proteins, mostly manufactured by the liver. They were originally numbered in the order of their discovery, traditionally using Roman numerals from I-XIII. Some of the numbers such as III and VI are not used any more and in recent years, many proteins that affect blood clotting have been discovered but have been given a name rather than a number. When a blood vessel wall is damaged, or any kind of wound occurs, a complex set of chemical reactions involving these coagulation factors (and acting rather like a waterfall) takes place.
The final step of the cascade of chemical reactions is to convert fibrinogen - Factor I - into fibrin, forming a mesh which clumps platelets and blood cells into a solid clot, plugging the hole and stopping the bleeding. Patients with Hemophilia A have deficient levels of Factor VIII, while patients with Hemophilia B have deficient levels of Factor IX.