Character Analysis
Slughorn seems so sweet and, well, ordinary when we first meet him disguised as an armchair (yes, our definition of ordinary is a little warped—we're dealing with wizards, after all). He just seems like a pleasant middle-aged armchair man who wants to avoid having to deal with Death Eaters, hence the disguise. He doesn't really make a strong impression either way.
Yet, Dumbledore has gone through a decent amount of trouble to visit Harry in London and bring him to meet Slughorn. That kinda implies that this guy is pretty important to something near and dear to Dumbledore's (and Harry's) heart, no?
We soon learn Dumbledore's game: it seems that Dumbledore really wants Slughorn to come back to Hogwarts as a Potions teacher, and Slughorn really doesn't want to come. However, when Dumbledore shows up with Harry, things change and suddenly he's all agreeable. Oh sure, Slughorn balks at first, but Dumbledore plays him like a magically-tuned violin to get what he wants:
SLUGHORN: ...but you're not leaving, are you?
DUMBLEDORE: Oh, I think I know a lost cause when I see one. Regrettable. I would have considered it a great personal triumph had you consented to return to Hogwarts. Ah well, you're like my friend Mr. Potter here. One of a kind. Well, bye-bye, Horace. Bye.
[Harry and Slughorn leave. Slughorn then chases them outside in his pajamas.]
SLUGHORN: Alright. I'll do it.
See what he did there? He knows Horace can't resist the lure of being near a celebrity as A-list as Harry, so he totally plays up how "one of a kind" the boy is.
But why is Dumbledore so keen to have Slughorn at Hogwarts? Well, to parse this out, we have to know a few things about Slughorn.
He's a Collector
When Harry first meets Slughorn at his home, he finds that the professor has a whole slew of photos of his gifted and/or otherwise famous former students. Kind of like a shrine to assist in his favorite hobby of name-dropping.
Apparently, having the best and the brightest around him was always super important to Slughorn, and that's why Dumbledore brought Harry around: he knew that Slughorn would just love the change to get close to "the Chosen One." As Dumbledore tells Harry, Slughorn basically likes to "collect" students, and Harry is a seriously amazing prize:
DUMBLEDORE: You are talented, famous, and powerful. Everything Horace values. Professor Slughorn's going to try to collect you, Harry. You would be his crowning jewel. That's why he's returning to Hogwarts. And it's crucial he should return.
Oh, and it totally works: after meeting Harry, Slughorn is totally ready to come back to Hogwarts. Step one, check.
Sluggy Did a Bad, Bad Thing
Dumbledore lets Harry in on the fact that he wants Slughorn back at Hogwarts, but he's not super forthcoming about the reason—well, at least not right away.
Eventually, though, Harry gets the truth: Slughorn taught Voldemort back when he was still Tom Riddle, and Dumbledore wants to know the deets about an important conversation that took place between them.
Why not ask Slughorn directly, you ask? Well, Dumbledore had tried that. In fact, he even had Slughorn give him a copy of the memory so that Dumbledore could view it in his Pensieve.
However, when Harry and Dumby view the memory, they can tell it's been altered. There's something about it that Slughorn doesn't want anyone to know...something he and Voldemort talked about. According to the version Slughorn gave Dumbledore, he shut young Voldemort down right away:
SLUGHORN: I beg your pardon? I don't know anything about such things and if I did, I wouldn't tell you. Now, get out of here at once, and don't let me ever catch you mentioning it again!
Such things? What are these mysterious "things"? We want to know more. Also we're pretty sure that's not the full story.
Dumbledore gives Harry the mission of getting the real memory from Slughorn, and with the help of a magic potion and his celebrity status, Harry succeeds. Before he spills the details, Slughorn begs Harry not to think poorly of him:
SLUGHORN: Please, don't think badly of me when you see it. You've no idea what he was like, even then.
We have to be clear here: Slughorn isn't an evil guy. He didn't mean to help Voldemort. He just...didn't really do anything to thwart Voldemort's interest in Dark magic. And then covered up the fact that he and Voldemort had ever talked about said Dark magic.
So, yeah, Slughorn is just a wee bit spineless, and his love of the gifted and famous kind of blinded him to how bad Voldemort really was...very Kato Kaelin-esque. But in the end, Sluggy does the right thing and gives Harry what he needs, so he gets some definite bonus points for that.