Compton Scattering Lowest Order Feyman Diagram I have some trouble understanding the lowest order feyman diagrams for compton scattering. 

Does two feyman diagrams mean that compton scattering can happen through two processes? What is the specific process in each case?
 A: Here is a clearer sum of the two lowest order diagrams:

It shows the two geometric ways energy and momentum can be exchanged between the two incoming particles to produce the two outgoing, to first order in a series expansion. 

Does two feyman diagrams mean that compton scattering can happen through two processes? What is the specific process in each case?

Crossections are calculated in quantum field theory as sums of a convergent series expansion of the original scattering amplitude. The series has diminishing constants in front of each order, as in all series expansions. These are the first order diagrams corresponding to two integrals over the available momentum and energy phase space. There is a one to one correspondence of all the elements in the feynman diagram to terms within the integral. Have a look here.
That there are two lowest order diagrams does not mean that they can be separated, they contribute to the calculations. The true value needs the sum of all orders, but for most experimental checks, the first order is sufficient. Here is how a calculation for these two diagrams goes.
