Instead of actually flushing (or pushing) the water down the drain, it just fills with water and drains out over the course of a few minutes. Anything liquid will drain out but nothing else (not even toilet paper). I want to know if there is something in the back of the toilet that I can adjust, because it seems like it doesn't even try to flush anything away.
Sounds like the flapper is leaking and/or broken altogether. See instructions on replacing a flapper here: http://www.wikihow.com/Fix-a-Flapper-on-a-Toilet
See other possibly useful answers here: How can I fix a toilet that flushes incompletely? and here: Why do my toilets flush slowly and often incompletely?
EDIT: Um, after re-reading the question, I wonder if the toilet isn't simply clogged. Does the tank re-fill and water stop flowing into the tank after flushing (you can hear when water is flowing)? If the tank re-fills and water shuts off, it isn't the flapper. If the tank re-fills and water stops, then it sounds more like the toilet is simply clogged. Did you try a plunger? If that does work and the tank does refill properly, then you'll need the drain snaked I think.
There are a number of ways to interpret what you are saying, but I suspect the other guesses are wrong. But let's check...
I'm assuming that you do not have a UK-style toilet. These are a whole 'nother kettle of fish, and the flush mechanism involves a syphon and black magic.
I'm assuming your toilet is NOT blocked. When you flush a blocked toilet, the bowl fills right up and then water slowly seeps away. This is fixed, as others have said, by plunging it. And not with one of those cruddy dome-shaped sink plungers, either! Make sure you get a toilet plunger. In fact, even if you don't have a blockage, if you don't have a toilet plunger, get one now. You will thank yourself when your toilet DOES block.
I'm assuming that your cistern fills up fully, so your flapper is not warped or leaking. That is, once the cistern at the back of the toilet reaches the "full" level, you can hear the running water stop, and if you flush at that point, your toilet still misbehaves.
With those assumptions out the way, let's get into TOILET PHYSICS! WHOO!
To get the toilet to flush, the water flowing out of the bowl needs to be enough to create a vacuum seal on the downpipe, and then the weight of the water falling sucks what's in the bowl out. But to fall so smoothly, the water needs some clear airflow... so there's a vent pipe that lets out onto your roof, that lets air in/out, allowing the water to flow freely, rather than having to let bubbles through and stuff.
So. Sounds to me like when you flush, either the cistern is not flushing fast enough, or the water isn't flowing fast enough from your toilet because either your drains need rodding, or your vent pipe is blocked (perhaps by leaves, snow, or even incompetent roofing contractors chopping it off, sealing it up, and roofing right over it! Yes, it happens!)
If it's your outlet or vent pipe, and plunging doesn't work, your best bet is to get a pro in.
But the cistern is easier. Pop it open, and look inside to see what happens when it's full and you pull the handle.
First and unlikeliest, if the chain from the handle to the flapper is too long, then it'll just lift a teensy bit and maybe not flap all the way open and so not let enough water through, fast enough, that it'll open all the way. the way flappers work is through buoyancy. They're filled with air, but they're tightly sealed all around, so there's no pressure underneath them to buoy them up. When they lift even a little bit, suddenly that water pressure's there, flowing under them and pushing them up... and the air that was in them bubbles out and is replaced by water! they sit there, out of the way of the hole they covered, until the water has all gone away, and the weight of the water they picked up pulls them down again, and they settle back down in place, pouring the water down into the pipe and filling with air ready for the next flush. So yeah, if the chain's not pulling the flapper up, shorten the chain.
If there just wasn't much water in the cistern in the first place, then you would have heard running water, either constantly, or occasionally (like every few hours) as water slowly seeped away through the bad flapper seal, and the cistern refilled itself. In that case, the flapper's bad, replace it.
If there wasn't much water in the cistern, but the water flow to it had turned off, then your stop valve needs adjusting to let more water in. If you've got a ballcock valve (ball on a stick!) then you can usually either bend the stick, or tweak a screw, to let the ball rise higher before it turns off. If you've got a "donut on a pole", then usually there's a rod to one side of the donut that has a little metal clip that can be adjusted that governs where the donut stops.
If there's something blocking the flapper from coming up, move the thing out the way.
If the flapper lifts up just fine, but the water flows out slowly anyway, then you've a blockage between cistern and toilet, probably limescale buildup in the pipes, or flakes of rust from the parts in the cistern, or gunk from tablets people have chucked in there, or some mixture of the three. Clean out any debris at the bottom of your cistern (it's good to do that anyway) and then try unclipping the flapper, and using a toilet plunger on the pipe there.
To me, this last sounds the most likely, from your description.
Just had this problem, and there's one more possibility I didn't see mentioned above...
Did you happen to notice any workers near your home in the hours before the toilet stopped working? Me neither. But it turns out they were down the street working and shut down the water main for their maintenance. When they shut it down and flushed the pipe, it created an air pocket in the system. Since the air pocket is blocked on both ends by the water Above and below the pocket (after the workers turned the main back on), it can't escape. Why is this a problem your shits should be concerned with?
When you flush, the air pocket creates back pressure which prevents the water in the bowl draining into the pipe (along with whatever unholy mess you left in there). I'm fact, you'll likely even see a little water splash up when you first flush (or in my case, a suspiciously large amount of water/urine mix splashing up and hitting you square in the face. Stop laughing. I'm about to fix your toilet and save you a few hundred on the plumber!)
How to fix?
This is a good news, bad news situation. The good news is hat it won't cost you a penny to fix. The bad news is that depending on what was in the bowl before you noticed the problem and how much water your bowl can hold before it overflows, you may need to break out the hazmat suit and remove the solids before you get started on the fix.
Solution: keep flushing.
Yes, that's right. Over and over again.
Every time you flush, you move the air pocket and allow a little air to escape. You can probably see where this is going. Wash, rinse and repeat until the air pocket is no more and the toilet starts flushing normally. Took about 25 flushes in my case. What can I say? Persistence pays off.
Disclaimer - You really only want to go down this road if everything else with your toilet is fine and you have already tried plunging like your life depended on it. A clog in the pipe is far more likely than an air pocket.