WARNING: This writeup contains spoilers. If you have not yet finished reading Harry Potter and the Half-Blood Prince, you probably should not read this.
Horace Slughorn is a wizard. He is quite old and had retired. He is not as old as Dumbledore, but he was teaching at Hogwarts when Tom Riddle was there, which was more than 50 years ago, so he is definitely old enough.
However, as we all know, Dumbledore can’t keep a Defense Against the Dark Arts teacher for more than a year, and as we learn in Harry Potter and the Half-Blood Prince, this situation has been going on a lot more than six years. So Dumbledore is getting pretty desperate to find new DADA teachers. In Harry Potter’s fifth year at Hogwarts, Dumbledore was unable to find anyone, so the Ministry of Magic appointed a teacher. Now it is Harry’s sixth year, and Dumbledore has finally given in to Snape and allowed him to switch from the potions master to the DADA master. So now Dumbledore must find a new potions master. He tracks down Professor Slughorn to ask him to return to his old post.
Although we never learn the details of how Dumbledore found him, it couldn’t have been easy, for Slughorn, afraid of Voldemort and the Death Eaters, has been changing residences weekly for over a year. Since Professor Slughorn is a man who likes his comforts, he has not allowed this to inconvenience him. He simply finds a home to his liking belonging to Muggles who are on vacation and moves in briefly.
Dumbledore wants him back, but Dumbledore knows he won’t want to come back. So Dumbledore brings that which Slughorn will not be able to resist - a celebrity. Yes, Professor Slughorn loves the rich and famous. He loves knowing them, and he loves creating them. How does he create them? During all his years as a professor at Hogwarts, he kept his eyes open for likely students - students who had the potential to go far, whether it was because of their family connections or their own talents. And he drew these students to him, holding parties to which he invited these elites. And he introduced them to others who could advance their careers. So it became a self-perpetuating and enlarging system. As more of his former students became the stars of the wizarding world, they became his contacts for the new stock he was bringing along.
But Slughorn has been retired for quite a while now, and there are dangerous things going on in the world. What celebrity would be enough of a lure to bring him back? Why, the greatest celebrity of the wizarding world, of course, the Boy Who Lived, the Chosen One, Harry Potter. Dumbledore brings Harry with him when he goes to Professor Slughorn’s current hiding spot, and it does the trick. Slughorn agrees to return.
But this is not all his significance to the story.
Professor Slughorn has a secret. Dumbledore has tried to get the secret from him with no success. You see, Tom Riddle was one of those bright, waxing stars in the Hogwarts firmament. Tom Riddle was one of those students that Professor Slughorn courted and groomed. He offered to help Tom get into the Ministry of Magic. And when Tom asked him a question, Professor Slughorn answered him.
Now Professor Slughorn realizes the consequences of his answering Tom’s question. He is ashamed that he is the one who gave this piece of information to the Dark Lord. And he does not want to admit it. How can Dumbledore break through his resistance and find out this vital piece of information? The same secret weapon he used before - Harry.
And what is Professor Slughorn's dark secret? He told Tom Riddle how to create a Horcrux.