First up, there's the classical pathway, which kicks into gear when an IgG or IgM antibody binds to its target antigen. When this occurs, a specific reactive site of the antibody is uncovered, and this area now binds to C1, the first protein of the complement system. This is all that is needed for the whole apparatus to get going, for the resulting cascade of amplifying complement factors culminate in a specific C3 convertase, which is an enzyme capable of splitting C3 into C3a and C3b (see above).
Next there's the lectin pathway. This is quite clever. A protein called mannan-binding lectin (MBL) is presently floating around in your blood stream, and is capable of recognising certain carbohydrate patterns characteristic of a wide variety of microorganisms. If it comes into contact with these patterns, it binds to them. This sets into motion another chain reaction: two proteins associated with MBL (MASP-I and MASP-II) then cleave C2 and C4, which then ... (you get the picture) ... until eventually the same C3 convertase as in the classical pathway is produced.
Lastly, there is the alternative pathway, so named because it was discovered after the classical pathway (although it is almost certainly older in evolutionary terms). This is a little bit more complicated than the other two pathways. It turns out that C3 doesn't exactly require a C3 convertase to be cleaved into C3a and C3b - this reaction is happening all the time, only at a rather slow and insignificant rate. The C3b can form a complex with another complement factor (B), which, after further modification, can become a C3 convertase (a different one to that the lectin and classical pathways produce, though). Thus C3 can spontaneously generate a catalyst that generates more C3, that generates more catalyst, that generates more C3 ... and so on in a rapidly amplifying loop:
Normally, however, this positive feedback loop is aborted by the fact that the generated C3 convertase is rather unstable, and so is quickly hydrolysed and inactivated by the water in our serum. But in the presence of certain molecules, including the surfaces of many bacteria, the C3 covertase becomes attached and stabilised from breakdown. This permits the above cycle to proceed unmolested, resulting in full activation of the complement system.
Those are the three starting points to activate the complement system. In the last post on this topic (for now, anyway!), we'll try to understand how important complement is by observing what happens when you don't have it.