Of the four girls whose backs I can see from my seat in world history, only one of them wears a properly fitting bra. When a bra doesn't fit, the whole world can see the problems it causes, and you'll likely wind up with a sore back. Sure the bra is evolved from a misogynist tool of the patriarchy, but these days it can make life a lot more comfortable for women with particularly large or floppy chests. That is if you find the proper fit.
A bra has three main components: the band, the cups, and the straps. The band is the primary source of support and therefore the most important part. It also seems to be the part women have the most trouble with. It should fit snugly but comfortably and remain horizontal at the level of the underbust all the way around. Most often women choose bands that are too tight causing unsightly (and uncomfortable!) bulges. I also see a lot of bands that ride up in the back, transferring the burden of support to the much weaker straps that the wearer has adjusted to be too short. The first row of images show the problems of an ill fitting band. Additionally, when buying a bra, choose one with a band that fits best at the last set of clasps so that as the elastic weakens and stretches over time you can adjust the fit by using the next set in.
Now the straps. These are really just there to help keep everything up and in place and aren't even really necessary--hence the possibility of a strapless bra. Never the less, they should fit well too. All bra straps are designed to be hugely adjustable but you will probably need help getting them just right unless your arms don't have any bones in them. If they cut into your shoulders they're too short, and if they slip off all the time they're way too long.
Measuring will help get you in the ballpark, but the best way to find a well fitting bra is to set aside a couple of hours and head to a big department store or a specialized store like Victoria's Secret where the employees are specifically trained to help you find the perfect fit. Expect to try on a ton of different size combos and styles to find the one that works for you. Finding the right bra takes effort, but the reduced potential for back problems and increased daily comfort are totally worth it.