I'm small-chested, so this is to add to what the larger ladies experiences are. (Although some of it is general boob-experience.)

My earliest experiences with my own boobs was the age of 10. I was climbing on some playground, and I slid (like I usually did as a child) down some high edge with my chest flat to it--and suddenly it hurt like fucking hell because the metal edge of the playground scraped against two small lumps that weren't there before. That was basically my first experience with mammary glands, and I expect it felt something like unexpectedly tapping/squishing your balls between your body and the metal edge of playground equipment.

When the boobs got a little bigger, there was a sort of fatty bag around that center "nut" of mammary gland. The fatty part of the boob just...feels like fat. It can get bigger or smaller as a woman gains or loses weight.

Normally the boob isn't more sensitive (for me) than other parts of my body, but I do try to shield my boobs from a direct hit on the gland inside because that hurts like hell. Since I'm a smaller-chested woman, I suspect I have less padding to protect it than larger-chested women do.

Nipples can get randomly sore from hormone changes (either period, or ovulation times of the month, or both). Particularly when I was a teen, mine would get excruciatingly sore for no obvious reason. I had a few times where I literally took some tape to put over the nipple so that the pain from just merely chafing against clothes would stop.

Running/jumping, the fat flops around and depending on things that can kind of hurt. If you're a guy with a big belly, try being naked and jumping. You'll feel a pull right around where the bottom of your belly tucks in above your pubic bone, where the weight is sort of stretching/straining the skin and tissues. Boobs flopping sort of feel like that, just higher on the chest. (Balls flopping MIGHT feel similar, but I'm not totally sure...the part where guys say they can feel roller coasters in their balls and the like isn't something I've experienced in my boobs, so there must be some sort of difference in sensation.)

You do have to learn how to accommodate boobs when picking stuff up, and moving your arms. The side of the boob can be wider than the natural width of your ribs, so if you move your arm forward it hits a bit of sideboob (or a lot if you're bigger). You learn to move your arm so it doesn't sideswipe your own boob. And you learn to move in other ways to accommodate your own body.

For example, if you want to tuck something under your arm, it's going to possibly poke your boob, so you have to decide how to get whatever you're carrying under your arm without poking your boob uncomfortably. And when you wear a messenger bag-type thing you might have to readjust the strap so it goes between the boobs instead of trapping one and strangling it. (Imagine the edge of your underwear unexpectedly trapping a stray ball in a weird position.) If you have a short waist, balancing something on your hip might require adjusting the boob at the same time.

Also, depending how big you are, you might not be able to lay face-down on a bed, or a floor, without putting some sort of support under your stomach or hips so you're not putting all your weight on your boobs. Boobs are in the way. They squish to some extent, but they don't squish into nothing, and the mammary gland is going to let you know, loudly, if you're compressing it.

Lastly, a lot of women wear the wrong sized bra. So if you're one of those women, you have an awful elastic thing around your chest all day, with edges that dig into your sides and your boobs. I mean, that's not really about the BOOB, but the experience of having them...if you have boobs, you've almost certainly worn the wrong-sized bra and know what it's like to have your boobs be uncomfortable during a 9-5 workday. (Then you come home from work and take off your crappy bra first thing.) If you're a guy and you've ever worn a knee brace or other elastic sports injury aid, bras are like that...tight, restrictive, supposed to do something, but possibly aren't the right size for you because "one size fits all" is bullshit.

Gonna point out /r/abrathatfits is an awesome, awesome sub doing great work for women.