I think Bob's details are completely correct, so there's not much to add on how Apollo actually worked.
I can add some thoughts on their design choices. Apollo's single-gas design has been much criticized, but I still think it was the right choice once the post-Apollo 1 pad safeguards were added.
Most of the Apollo missions involved a lot of EVAs, and all but two carried a LM. It would have been completely impractical to operate the LM with an air-like mixture. It would have to be much heavier to withstand the increased pressure, the life support system would be more complex and heavier, and the astronauts would have to decompress before every EVA as they do on Shuttle and ISS. And as everybody knows, weight constraints on the LM were extreme. Since the LM was docked with the CSM on the trip out, this forced the CM to use a single gas atmosphere as well. (Apollo did eventually mate with a spacecraft using air during ASTP. It carried a complex "docking module" airlock and gas supply system to make the two compatible.)
The only part of the post-Apollo 1 design I might change is the choice of gas mixture on the pad. As Bob explained, they used 60% O2, 40% N2 to maintain a sufficient ppO2 once they reached orbit at a cabin pressure of 5 psia. A 60% O2 atmosphere is a lot safer than 100% O2, but it's still obviously worse than air. So one idea I had is to launch with ordinary air (at +2 psig to keep the cabin pressurized) and instead of bleeding down to 5 psia and holding during ascent, bleed all the way down to a vacuum. The astronauts' pressure suits, which always contain pure O2, are simultaneously bled down to their nominal 3.75 psia and held there. Then the cabin is repressurized with pure O2 and you're done. No need to slowly purge N2 from the cabin over a few days, which undoubtedly wasted a lot of O2.
I mentioned this idea to Owen Garriott a few years ago and he couldn't see anything wrong with it. Then I saw two possible problems. As flown, suit pressure decreased from about 17 psia (+2 psig at sea level) to about 5 psia rather quickly during ascent. All the N2 had already been purged from the astronauts' bodies before launch, so this evidently did not risk decompression illness. My plan would bleed them down from 17 psia to only 3.75 psia, and this greater change would have to be studied to ensure no decompression problems. A second problem would be that the suits would become stiff once the cabin bled below 3.75 psia during ascent, and this might interfere with their ability to operate the controls. This could be avoided by holding the cabin at 5 psia during ascent, delaying the vacuum purge until they were in orbit.
Owen didn't think it was possible to get the bends on pure O2, but I'm not so sure. It is even more soluble in blood than N2, although the fact you can metabolize it means you can get rid of it without having to breathe it back out.