Right, so, biopower. I don't mean, like, ethanol fuel; I'm not talking literally about energy. I'm talking about the other kind of power. You know - power. Control. The ability to make things happen the way you want them to happen. And biopower is power over bodies. That's the "bio-" in biopower, clue is in the title.
It used to be, back in the day when there were kings and shit, they had power over your death - if you were a problem they would just fucking kill you, because fuck you I'm the king and I can do that. And that was more or less it. "Do what I say, because I say so. Just don't give me any reason to kill you, and everything will be fine." What were those things? Pay your taxes, worship the right God, don't attempt an uprising, and go fight places we need you to fight. In other words, things that kept the state in place and secure. Sometimes good kings would, I don't know, feed their people, but that was just so they didn't have hungry peasants all burning cottages and shit. As long as the king wasn't threatened, he didn't give two fucks about "it's so cold" or "clean drinking water" or "seriously we're hardly getting by on like 80 hours of farming a week and we could really use some ditches dug." Fuck whatever rabble his whiny subjects happened to be rousing at the time.
But eventually, kings began to figure out they could use power to create power. And the more we learned, the more the law could be used to create rather than destroy. With the advances in science and genetics and the discovery of evolution, we knew more about humanity than ever before, and slowly, power over death got replaced with power over life. The state stopped trying to just survive - now states are actively trying to create a healthy, productive populace. And you can't create a healthy, productive populace just by fucking menacing them with the death penalty all the time, not that some real assholes didn't try that route. No, now laws do a whole lot more than just condemn shit the state doesn't like - they decide what is normal and healthy and okay for people to do.
So what's the problem? "Healthier, happier people? More caring state? This sounds great!" And that's exactly the fucking problem. That sounds great, but what exactly is healthier and happier? Who gets to decide that? You? Your buddy Steve? Fuck you, and fuck Steve. Biopower is literally power over your body, my body, er'rybody's body; the state gets involved in everything - health, labor, education, your goddamn sex life, and it tells you what you can and can't, should and shouldn't do, and it backs it up with guns. And if you're all, "Hey, uh, I don't feel more healthy and productive," well there's an easy answer for that! "Oh, you don't? Hm... well, uh, how about fuck you. You just have the wrong ideas. Glad that's all cleared up!"
And biopower doesn't stop there - it's also power over the populace as a whole. The state doesn't want to just make you healthy, it wants to make us healthy. Oh, and hey, by the way, it gets to decide who "us" is. Hitler never could have pulled off the Holocaust back when rulers just had to "preserve the state." But as soon as he was all, "Oh, I'm doing it for the health of the nation! We have to clean things up! Make Germans better!" that's when the whole country jumped on board. He didn't pretend to be not racist. It was like the centerpiece of his policies. He just made racism sound like something the country had to do for its own good. And who doesn't want to do "good" for their country, or for humanity?
And now maybe you're thinking, "Oh! That's terrible! We have to stop electing people who abuse the power of the state!" Yeah, it's not that simple. What counts as 'abusing' power? What are the 'right' ideas? It hasn't even been fifty fucking years since "homosexuality" stopped being called a mental disorder in America. Who the fuck are you going to elect in 1960 to help homosexuals become "healthy and productive?" More importantly, who knew back then that the state was being abusive?
If, tomorrow, some crazy lunatic got on TV and started ranting about how juggling is destroying America and fuck juggling, and enough people believed him, we would start electing politicians who promised us they would get jugglers off the streets, man, and make them do something productive for once. And then in fifty years, someone would come along and say, "Why can't we juggle?" And everyone would scoff and go, "Look at this fucking guy! Wants to juggle. Hey what are you, some kind of juggler? Huh?" And that's bad enough, but then we'd throw him in jail or a mental asylum, thanks to biopower, because we want to protect ourselves from that crazy juggler guy. That's super important here: we would be the ones putting him in jail. Sure, "the state" would do it, but because we elected a state that was anti-juggling and gave it the power to do things like imprison jugglers.
So the problem isn't just choosing the right kind of state; the problem is that we don't even know what "the right kind of state" is. We all have these ideas of right and wrong and normal and we choose a state according to those, and sometimes we fuck up what is right and wrong and normal. And when the state has the power to literally control what citizens do with their bodies, and we justify that power with "it's for your own good! and for everybody's good." There is no limit to what the state can justify, and if you don't like it, you must not want to be healthy. You must be broken. And you know what we can do with people who are broken, thanks to biopower? We can "fix" them. We can "fix" them by deciding what they learn and eat and do for a living and whatever else we want.
At bottom, biopower is what you get when human well-being suddenly becomes subject to the political whims of an imperfectly informed society, which (it turns out) is the only kind of society.
The Wikipedia page about biopower is, as of this writing, fucking terrible. So don't bother there, unless you're interested in the original lectures in which Foucault lays out the idea of biopower.
He first publishes about biopower in The History of Sexuality, Vol. 1, though he discusses it only briefly in part 5 of that volume.