At least from the movies, we can gather that all the students and faculty at Hogwarts are human, but the Potterverse is full of other magical creatures, some of which interact peacefully with humans. Has any other race of beings ever attended any of the Wizarding schools around the world? If so, who, where, and when?
There are a few examples of students that are not full-blood human attending a school of magic.
Half-giant, attended Hogwarts.
Quarter-Veela, attended Beauxbatons. We can safely assume her sister, Gabrielle Delacour, also attended Beauxbatons.
Part-goblin (several generations removed), attended Hogwarts
If you move beyond students to school staff as well, you can include:
Centuar, Divination teach at Hogwarts. The only non-part-human on this list!
Half-giant, headmistress at Beauxbatons.
Several people have asked if Remus Lupin, a werewolf, should be on this list. He doesn't count because a werewolf is a human suffering from a condition called lycanthropy. A werewolf is a human, just as a person with cancer is a human.
Another situation worth mentioning is the ghost Professor Binns. Does a human cease to be a human after he dies? Does he transform from a human being to a human been?
They can’t in Britain - they can’t get wands.
In Britain at least, it’s very unlikely that any pure non-human (rather than a part-non-human, like Hagrid) could attend Hogwarts. All students are required to have purchased a wand before they start at Hogwarts.
- Harry Potter and the Philosopher's Stone, Chapter 5 (Diagon Alley)
No non-human creatures are allowed to use or carry a wand, so they wouldn’t be able to own and use one to attend Hogwarts with.
“Come off it, Amos,’ said Mr Weasley quietly, ‘you don’t seriously think it was the elf? The Dark Mark’s a wizard’s sign. It requires a wand.’
‘Yeah,’ said Mr Diggory, ‘and she had a wand.’
‘What?’ said Mr Weasley.
‘Here, look.’ Mr Diggory held up a wand and showed it to Mr Weasley. ‘Had it in her hand. So that’s clause three of the Code of Wand Use broken for a start. No non-human creature is permitted to carry or use a wand.”
- Harry Potter and the Goblet of Fire, Chapter 9 (The Dark Mark)
It’s unclear if this law is in effect in other countries as well or if it’s specifically a British law, but certainly in Britain, purely non-human creatures wouldn’t be able to attend Hogwarts, since they’d require a wand but it’s illegal for them to have or use a wand. This may be different in other countries, though, so it’s still possible that purely non-human creatures have attended some of the other wizarding schools.
Possibly Voldemort, according to Hagrid.
According to Hagrid in Chapter Four of Philosopher's Stone, Voldemort might not be human:
"Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die.
It's unclear exactly when Voldemort would have become non-human. He had already begun using Dark Magic, and he had already murdered, while yet a student. Dumbledore in Chapter Twenty-Three of Half-Blood Prince seems to have associated the dehumanization with creating Horcruxes:
And I thought I knew what that meant, though the Death Eaters did not. He was referring to his Horcruxes, Horcruxes in the plural, Harry, which I don’t believe any other wizard has ever had. Yet it fitted: Lord Voldomort has seemed to grow less human with the passing years, and the transformation he had undergone seemed to me to be only explainable if his soul was mutilated beyond the realms of what we might call 'usual evil'..."
In any case, by the time of Philosopher's Stone he was certainly already at the point Hagrid referred to, and he then spent a year attending Hogwarts attached to Professor Quirrel.
Possibly Quirrel, according to Harry.
In Chapter Eight of Prisoner of Azkaban we have the following passage (my emphasis):
Harry stared. Not only was this the last answer he'd expected, but Lupin had said Voldemort's name. The only person Harry had ever heard say the name aloud (apart from himself) was Professor Dumbledore.
Yet in Chapter Seventeen of Phlosopher's Stone Quirrel says "Voldemort" a couple of times:
Tried to frighten me — as though he could, when I had Lord Voldemort on my side...."
Lord Voldemort showed me how wrong I was.
Perhaps, then, Quirrel was not a person (perhaps because he was hosting Voldemort), yet he attended Hogwarts.
(To be fair, Harry also heard Hagrid say "Voldemort" in Philosopher's Stone, but Hagrid is already mentioned in another answer.)