Blood is a tissue made up of many components, and the coagulation process is a complex one. Most often people focus on the cascades of proteins that,at the location of a cut, form fibers to help clotting. Like spiders forming silk,
Aspirin works because the process also involves a type of cell known as a platelet. These are made by the body just to aid in clotting. This is what Aspirin works on, there are other drugs that work on them as well. Platelets are normally smooth flat cells, kept that way by the presence of nitric oxide produced by the walls of blood vessels.
Where the vessels ate torn, nitric oxide is not produced. The platelets shoot out extensions similar to the burr of briar plant. Form following function - just the perfect for any sort of fiber to stick to.
Normally, the protein fibers glom on to the activated platelets as part of making a clot. Sort of like adding gravel to plain cement. The cement has something to take up volume and help it hold together. Another example would like the platelets being the pylons of a the proteins being the steel cables that connect them.
Take away the pylons, and you can only make a short bridge.
Anti-platelet agent - drug that inhibits the ability of platelets to do their job.
Anti - coagulant - any drug that inhibits the ability of blood to clot.
The layman term for them is "blood-thinners"