Ok, hopefully this isn't too upsetting. I made it look a little like a football play diagram. Sorry about all the pink.
What we're talking about with both the pill and the IUD are two hormones either individually or in combination, estrogen and progestin (and also, in the case of some IUDs, copper, which I just learned tonight - see the FAQ link above). So a woman might take a mix of estrogen and progestin or else they do a progestin-only program - sometimes because the estrogen reacts poorly for them.
From what I've read, the progestin is the real root of the matter in terms of where anti-abortion one-upsmanship crosses into birth control territory. One of the effects of progestin is the thinning of the lining of the uterus, making for poor conditions for implantation. So the freshly fertilized egg, already a baby according to some GOP candidates, doesn't stick and grow, it just passes on out. Or it might stick, but just not take and pass out anyway. Of course, you don't need progestin to make any of this happen. A fertilized egg can just not implant sometimes and the woman might not even realize it. And, something I've come to learn anecdotally, it's not at all uncommon for the uterus to shed its lining in the early stages of pregnancy anyway (before I knew this I generally thought of a miscarriage as an unsuccessful birth).
It's also worth knowing that the pill isn't just about birth control. Because the progestin thins the lining of the uterus, that can make a woman's period lighter. (Less stuff to pass I guess?) So a woman may take the pill for reasons that have as much to do with her period as preventing birth (see, for example, dysmenorrhea [not as gross as it sounds]). Furthermore, as a woman gets older, she'll sometimes need to take hormones just to keep her body's levels in the right balance. I'm not just talking about menopause, I'm talking about the time leading up to that too - perimenopause. So a woman may be fertile (needing birth control) but also hormonally irregular (in need of hormone supplements). My point here is that we keep using the term "birth control" as though the only purpose is so we can boff away without fear of pregnancy, but for many women the pill is medicine.
Obviously the TRMS man cave is a relatively inexpert place to find gynecological information, so I appreciate any corrections and clarifications you can offer to the material presented here. I've even saved the psd file of the graphic in case changes are warranted.