I see "Darwinism" as a bit of a vague term, not really clear. I guess all of Biology now is Darwinist, kind of like all of physics is Einsteinian...but maybe you are referring to Darwin's original theory, as opposed to how much it has changed in the present day?
"Natural selection is one important piece of the Theory of Evolution (ToE.) Basically there are two steps in change of populations over time: random variation in offspring, then natural selection.
An analogy might be if we were going to try to design an airplane wing. (Engineers actually did this.) They program the computer to choose whatever design is most aerodynamic. The computer throws some numbers in and throws out all kinds of crappy designs, square ones, ones with doo-hickeys, etc. Then the algorithm SELECTS the most aerodynamic, and throws it back into the process. The computer spins around again and spits out another bunch of designs from there, then it selects the most aerodynamic. After a few rounds of this process, they got a wing better than they could have designed. It has a random piece, and a selection piece. ToE says this is something like how organisms get "designed" over time, and explains why they look designed, without a designer specifying each individual species.
I'll leave your last question to PW or someone more educated for now, because the answer is A LOT. There's this whole period during which ToE and genetics were separated, and the relationship not understood. Then they came to together in the Modern Synthesis, which is important to the history of science.
Some other terms that come to mind that are important are punctuated equilibrium (aka punk eek), genetic drift and evolutionary development, or evo devo. If no one comes along with good explanations, I'll try to supply when I have more time.