I'm convinced the Time-Turners were used to study time by the Unspeakables in the Department of Mysteries, as one of the prevailing forces of the universe that cannot easily be understood, manipulated, recreated, or tampered with, without severe ramifications. From a logical standpoint, I don't think in a million years (no pun intended!) would the Ministry of Magic have agreed to allow a student to possess a Time-Turner, no matter how highly recommended that student came. Look at Hermione -- even she used the Time-Turner outside the parameters of her agreement (to use it only for attending extra classes). DVK is right in that there are no canon examples of anyone using a Time-Turner, aside from Hermione.
The Department of Mysteries is complicated, obviously, and it studies nebulous topics: how thoughts can be as dangerous as weapons (the Brain Room, where Ron is injured by the brains with tentacles); how love or coercion under the guise of love can induce desperation, foolishness, and irrational devotion (Merope Gaunt and Tom Riddle Sr.); the Veil room where that most ultimate of mysteries is studied, death (Harry and Luna hearing voices beyond the veil; Thestrals; perseverance after loss; fear of death (Horcruxes)). And then there was the Time Room, where we can guess what is studied. There we find the bell jar with the egg inside that repeatedly hatches and then unhatches (the one that later gets caught on the Death Eater's head and causes him to loop from baby head to adult head, over and over), in an endless loop and the cabinet of Time-Turners on the wall, that keep falling, smashing, repairing themselves, and repeating this over and over again would indicate that the Unspeakables engaged in time travel at the very least for research purposes.
Sebastian Redi of Austria makes interesting points in his essay Time Travel in Harry Potter, hosted at The Sugar Quill. He identifies different types of time travel that might be relevant in Potterverse and may be some of the types of time travel the Unspeakables engage/have engaged in.
Open Single Flow: The open single flow model is the easiest when there is no time travel, but the most complicated when there is. In the OSF model there is a single thread of time which is spun along as people make decisions. The model offers completely free will. The existence of definite prophecies requires a relaxation of the model, allowing for single fixed points in the thread. A time traveller going back can easily create a paradox by making a modification that prevents his travel. Back to the Future employs a variant of this model, the Bendable Open Single Flow. Paradoxes can be created, but they need to be resolved. The thread of time constantly seeks to repair itself. Damage inflicted by time travellers will only slowly take effect as repair of the damage gradually gets more unlikely. But still, the OSF does not lend itself to time travel, even though it is the most intuitive model. Forward time travel ought not to be possible in this model because the future doesn't exist yet.
Open Multiple Flow: Here we enter the realm of parallel, or rather intersecting, dimensions. The OMF model spawns a new reality for each and every possibility. Every possible world is a reality of its own. Strictly seen, there is no free will in this model. You cannot choose, you always do everything. If you have a choice, you'll spawn as many worlds as you have possibilities, choosing a different one in each world. The less likely you were to choose a particular possibility, the less real the world is.
The core problem with this model is whether there is a main world and which it is. Is one world more important, more real than the others? Why is it this particular world? If it is the most likely one, how come that unlikely things happen to us? Wouldn't that mean that we don't live in the main world? Terminator employs such a model. Many, however, believe that Harry Potter uses a mixture between the Open Single Flow and the Open Multiple Flow [models of time travel].
Open Repeated Flow: Essentially it is a large modification of the Open Single Flow, introducing some Open Multiple Flow traits. Again time is a single thread, but time travel can modify that. A time traveller is able to cut off a finished part of the thread and reweave it. in the future, somebody travels back. By doing this, he cuts of the existing thread at the point of his arrival [in the past] and lays it aside, its only purpose being to provide existence to the time traveller and its existence only maintained by the memory of the traveller. The thread will begin to reweave itself, but this time there's the time traveller in it, equipped with knowledge of the future and capable of modifying it. The Grandfather Paradox, for example, is solved. I can go back in time to the point my grandfather lived. By doing so I cut off the entire part of the thread in which my father and I live and lay it aside. It provides existence there for me. I then go on to reweave a new thread. I kill my grandfather. In the new thread, neither my father nor I appear, but that doesn't matter. I'm there, come over from the cut piece of thread. Nobody in the real world has any memories of the events in the other piece of thread, of course, except me. There is full free will in this model.
Closed Single Flow: In Closed Single Flow, you see a single completed thread and an ant, the Now, crawling along the thread, representing the passing time. In such a model, everything is predetermined. Prophecies pose no problem at all, they're merely glimpses of different parts of the thread. Free will is an illusion. The characters don't know it, but every single one of their choices is already predetermined. This model is unattractive to many because of this. They don't want free will to be an illusion. But free will in a predetermined world? Since I don't know about what will happen, my choices seem to be free and seem to matter, and that's all I ask. The nice thing about Closed Single Flow is that time loops don't matter. It is already predetermined that in the future I will come back to save my current self, and thus there is no question of how I am capable of travelling to save me in the first place.
The author concludes that Prisoner of Azkaban falls under the Closed Single Flow model of time-travel. Personally, I think Prisoner of Azkaban represents an extremely clever use of time-travel and is done very well.
What were the Time-Turners used for? I presume that Hermione was not the only witch or wizard to get a bit adventurous with a Time-Turner over the history of Time-Turners' existence. So there's the obvious: they were used for time-travel. I think, thought, deeper than that is why they were used for time-travel -- what purpose did the Department of Mysteries have for travelling through time -- did they change history in any way? I also think that the Time-Turners are part of a group of time-related studies, which I already mentioned, because one of the themes in Harry Potter is how time is infinite; we are here for a very brief stay in the grand scheme of things; how do we make the most of the time we're given? If the opportunity arose to travel back into time, what choices would we make? Choice is a major theme in Harry Potter too: It is our choices, Harry, that show who we truly are, far more than our abilities. (Albus Dumbledore - Chamber of Secrets). Time is life and life is choices.
Yes, I think Time-Turners represent the question: what do we do with the time that we have (and we'd better get it right the first time around, because, unlike in Harry Potter, there are no re-dos.) It kind of goes back to that quote from Strictly Ballroom: A life lived in fear is a life half-lived. Harry certainly didn't let fear rule him and I think a strong theme in Harry Potter is doing what is right and not what is easy in the face of adversity, and we only have one chance to make a decision. After that, it's done. One thing seems to be clear, though. The Ministry of Magic doesn't seem to use the Time-Turners to alter bad things to come, wuch as Grindelwald and Voldemort. Of course I have no canon verification of this; I'm just trying to extrapolate logically.