In the beginning of Half-Blood Prince we see Dumbledore recruit a former teacher of Hogwarts — Professor Slughorn. A potions teacher. It is presumed that he was teaching in the past, and left at some point.
Do we have canonical answer explaining how, and why he stopped teaching?
This is addressed in Short Stories from Hogwarts of Power, Politics and Pesky Poltergeists. In brief, Slughorn became increasingly disturbed by his part in Voldemort's rise to power, specifically his famed immortality. When it became apparent that Dumbledore wouldn't stop in his attempts to interrogate him about his interaction with Tom Riddle (and hence reveal his most guilty secret, that he'd helped him to make a Horcrux to prevent his death) he chose to retire, outwardly claiming failing reflexes and 50 years of loyal service to the school.
When the wizarding world fell into war, and rumours swirled that Voldemort had, somehow, made himself immortal, Slughorn was sure that it was he who had made Voldemort invincible, by teaching him about Horcruxes (this guilt was misplaced, as Riddle already knew how to make a Horcrux, and had feigned innocence in order to find out what might happen if a wizard made more than one). Slughorn became ill with guilt and fright. Albus Dumbledore, now Headmaster, treated his colleague with particular kindness at this time, which had the paradoxical effect of increasing Slughorn’s guilt, reinforcing his determination never to tell a living soul what a dreadful mistake he had made.
Lord Voldemort made no attempt to seize Hogwarts on his first ascent to power. Slughorn believed, correctly, that he was safest remaining in his post rather than risking the outside world while Voldemort was at large. When Voldemort met his match upon attacking the infant Harry Potter, Slughorn was even more jubilant than most of the wizarding population. If Voldemort had been killed, Slughorn reasoned, then he could not have made a Horcrux, which meant that he, Slughorn, was innocent after all. It was Slughorn’s extremity of relief, and the disjointed phrases he let fall in the first rush of emotion after hearing of Voldemort’s defeat, that first alerted Dumbledore to the possibility that Slughorn had shared Dark secrets with Tom Riddle. Dumbledore’s gentle attempts to question Slughorn, however, caused him to clam up. A few days later, Slughorn (who had now completed a half century of service to the school) tendered his resignation.
Throughout Half-Blood Prince he is described as coming out of "retirement". Specifically, when Harry and Dumbledore go to convince him to come back, he specifically mentions his age/weakness (my emphasis):
"Well, how have you been keeping, Horace?" Dumbledore asked.
"Not so well," said Slughorn at once. "Weak chest. Wheezy. Rheumatism too. Can’t move like I used to. Well, that’s to be expected. Old age. Fatigue."
"And yet you must have moved fairly quickly to prepare such a welcome for us at such short notice,' said Dumbledore. ‘You can’t have had more than three minutes' warning?"
Slughorn said, half-irritably, half-proudly, "Two. Didn’t hear my Intruder Charm go off, I was taking a bath. Still," he added sternly, seeming to pull himself back together again, "the fact remains that I’m an old man, Albus. A tired old man who’s earned the right to a quiet life and a few creature comforts."
And:
"What would the Death Eaters want with a poor broken-down old buffer like me?" demanded Slughorn.
We can easily figure out that Slughorn is no youngster. He was already teaching when Voldemort was in school, and we know from Chamber of Secrets that Voldemort is about 54 years older than Harry. So at the time of Half-Blood Prince Slughorn is likely well over 90 years old. It is thus, perfectly reasonable that he would have retired at some earlier point due to his age/declining health. In fact, he even insinuates that Dumbledore should think about retiring for that reason as well:
"You’re not yet as old as I am, Horace," said Dumbledore.
"Well, maybe you ought to think about retirement yourself," said Slughorn bluntly. His pale gooseberry eyes had found Dumbledore’s injured hand. ‘Reactions not what they were, I see."
