We left our spunky little Foundation steadily gaining in regional power, but in the context of the larger galaxy, that power is confined to a few barbaric, outlying planets that nobody gives a whiff about anyways. The Galactic Empire is still technically in existence, although outer planets all over are declaring independence and slacking off on their taxes, and the empire's power is being eaten away at. Trantor is still technically in control of the galaxy's tax forms.
Now, when writing about a bunch of books in a row, I can't really discuss 2-4 without first talking about what happened in 1-3, so these reviews are not a good thing to read if you don't want to know what happens.
Part One: Yadda, Yadda, Yadda
Bel Riose, one of the emperor's generals, does not like the threat, small though it is now, that the Foundation poses to the empire, and he decides to go rogue and attack it. The emperor finds out and believes Riose to be a bigger threat to the empire than some backwaters planet, and has him executed. The people of Terminus grow steadily complacent, putting blind faith in the Seldon Plan, and thinking that their power will inevitably grow and grow according to plan. They forget that the early Seldon Crises were averted not because of destiny, but because key people noticed the upcoming threats and took decisive action to prevent ruination.
Part Two: The Mule!
100 years later. Trantor's been sacked, the Galactic Empire is like cosmic dust in the wind, the Foundation is largest Galactic power, all because of their economic reach (no invasions of planets for them!), and the Seldon Crisis that was foreseen (a bunch of rich trading planets revolt against the Foundation and try to secede) is ignored because of a much larger, unforeseen threat: THE MULE!
The Mule is highly secretive - nobody knows his real name, almost nobody has ever seen him, but the rumors abound. Rumors of his massive strength, and fire-eyes, and so on. The reality is that his strength comes from his mind: he can reach into people's minds and permanently adjust their emotions and loyalties so that they're more conducive to the Mule's tastes, which could entail loving him, fearing him, feeling helpless in the face of his attack, etc.
He uses this unique, freak ability to quickly climb to power in planets near Terminus, and plans to face off against the Foundation. The men in charge assume that this is yet another Seldon Crisis (yaaawn) and that one of Seldon's patented videotapes/holograms will appear and tell them exactly what to do to beat the Mule. Instead, Seldon pops up and warns them about the secessionary trading planets. Oh shit.
Meanwhile, on those trading planets, Toran and Bayta Darell sneak over to Kalgan, where the Mule has set himself up real nice, and do some spying. With the help of Han Pritcher, an insubordinate Foundation captain, they leave the planet in haste, and harboring a fugitive: the Mule's jester, Magnifico Gigantico. He hasn't done anything wrong, really, he just wanted to escape the Mule's vicious grasp. They pick up another guy on the way, Ebling Mis, who's a psychologist with the Foundation, and they all go off searching for the Second Foundation.
Hari Seldon had talked about two foundations back in the days when he was alive, the second being formed "at star's end," but hundreds of years of searching had proved futile, and everybody assumed Seldon was talking crazy, or trying to mislead certain people, or something else deviously clever. These guys figure that if they have a shot at finding the answer to the Second Foundation riddle, they should go to the giant library on Trantor, which was the only part of the planet that was left untouched in the sack by barbarian hordes (students at the university protected it), and is still the largest storehouse of galactic knowledge. The Second Foundation might be further along in their development of physical weapons, or they might have other weapons that could be used against the Mule's unique mental powers. Last hope for the universe, and all that.
Mis puts his nose to the grindstone when they get to the library, becoming completely focused on trying to figure out what Seldon could have meant, and where the Second Foundation could be located. He loses hair, he loses weight, and finally, finally, he figures it out. He's literally on his deathbed when he calls in the Darells (Magnifico was already with him, having been his helper) and tells them the good news. He opens his mouth to announce the location and BAM! Bayta shoots his fucking head off.
She has her reasons. She realized that Magnifico, the sad clown, was actually the Mule, who was using them both to gain access to broad swaths of people to be influenced and to find out where the Second Foundation was so he could destroy it. When Bayta figured this out, she knew that at all costs, the Mule could not find out what Mis had discovered. So she killed him dead. (He was dying anyways, because the Mule has placed so much mental pressure on him.) The Mule goes back to his empire in which all of his subjects have been made to believe that they love him, and Bayta and whatshisface go... do something.
The upshot is that they, and we, the readers, know that Second Foundation does exist, and instead of building political, scientific, and economic power, it was staffed with psychohistorians and tasked with studying "mentalics," i.e., what the Mule has innate control over.