There is no breakdown in any canon materials of how exactly Time-Turners work and what the limitations are. But! That will not stop me from discussing what I believe to be reasonable rules and guidelines, and I also believe that my reading of these things is supported by the text.
There are two major questions about Time-Turners that immediately come to my mind. First, can they really change the past? Second, how is one's age affected by Time-Turner use?
Can Time-Turners Change the Past?
Some of my favorite fics involve the use of Time-Turners to change the past. It's a wonderful plot device and I love it to bits, especially in the gift fic written for me for sirry_slash Secret Santa exchange, This Time, by anonymous (rated NC-17). That said, however, in canon...
You cannot change the past using a Time-Turner.
I feel confident asserting this, based on the extensive use we see at the end of Prisoner of Azkaban. As discussed in the Wikipedia article on Time-Turners, everything Harry and Hermione did when they "went back" was already done the first time around. I could go through this item by item, but I'm sure it's been done somewhere. I will instead just give you the first one.
When Hermione turns the Time-Turner, she and Harry go back to where they were earlier in the evening, the entrance hall.
He was standing next to Hermione in the deserted entrance hall and a stream of golden sunlight was falling across the paved floor from the open front doors. He looked wildly around at Hermione, the chain of the hourglass cutting into his neck.
"Hermione, what -?"
"In here!" Hermione seized Harry's arm and dragged him across the hall to the door of a broom closet; she opened it, pushed him inside among the buckets and mops, then slammed the door behind them.
[Sorry, no page numbers.] Now, flip back a few pages/chapters in the book to what happened the first time Harry, Ron, and Hermione were going through the same moments in time.
They went down to dinner with everybody else, but did not return to Gryffindor Tower afterward. Harry had the cloak hidden down tie front of his robes; he had to keep his arms folded to hide the lump. They skulked in an empty chamber off the entrance hall, listening, until they were sure it was deserted. They heard a last pair of people hurrying across the hall and a door slamming. Hermione poked her head around the door.
That "last pair of people" were the future Harry and Hermione. This and a dozen other little things show us that the histories fit together like a puzzle - nothing is changed when they go back.
I still love to read fics where it does change the past, but it's also possible to write a fic where everything fits neatly together so that it's all the same as it was the first time around. That's what I like to do in my Time-Turner fics, although it is definitely a challenge.
How Old Is Hermione?
This question is a big puzzle to me. I see three main options. When you put a Time-Turner around your neck and go back one hour in time, either:
1. You continue to age normally, so that you are an hour older than you should be.
2. You de-age when you use it, so that your age matches that of the original you.
3. Your age doesn't change, but you are magically prevented from aging until you get to the point in time where you used the Time-Turner.
I think any of these could be the case, but each has some interesting implications.
In option 1, this would mean that Hermione gained a whole lot of hours during her third year. I don't have her class schedule on hand, but didn't she take four or five extra classes? Even if they met once a week for one hour each, five extra hours a week for an entire school year makes her days older than she should have been. Okay, maybe a few days older isn't a big deal, but still, it's something to keep in mind.
De-aging is a canon possibility as far as I'm concerned, thanks to that glass bell jar in the Department of Mysteries and the implications of an Aging Potion. This would also make it possible for characters to go back from adulthood to teenager (for example) and still be able to pass as themselves. But this option is limiting, since you can't go farther back in time than your own lifetime (and, practically speaking, not even that far, or you'd be helpless).
The third option is probably the most convenient from a story-telling point of view - it takes out all the problems of characters being ages they shouldn't be, and still lets you take a character back a hundred years or whatever. But if this sort of magic is possible, then it has implications for the rest of the wizarding world - perhaps there is a potion that replicates this effect, so that no one has to suffer the effects of aging. Problematic? I think so.
Out of these three, I think the second one is probably the most likely. This would be consistent with the main use of the Time-Turners being things like going to two meetings or classes at once, being in two places over the course of a single evening. I don't think the Time-Turners can be used for serious time travel, especially since there doesn't seem to be a way back to the "present." Anyway, this would mean that Hermione is just the right age.
Thoughts and opinions very welcome!