Fortunately, not much can go wrong with your toilet. Your toilet can give you trouble only in two ways - either it will be clogged or it will have developed some leaks.
Now leaks can develop anywhere in your toilet but the common places for it are (a) the toilet seal on the floor where it is planted, (b) the flush tank, (c) water supply pipe to the flush tank (d) water leakage from flush tank into the bowl, and (e) overflowing inside the flush tank.
The toilet seal
The bowl has a non-permeable seal, generally made from wax, which prevents water on the floor to seep into the floor where the bowl is planted. To fix this, follow this procedure
Turn off the water supply to the flush tank, empty the flush tank.
Empty the bowl water through a sponge, Unfasten the t-bolts that hold the bowl to the floor.
Unfasten the water supply pipe from flush tank to the bowl, lift and turn the bowl upside down next to you.
Remove the previous seal on the bowl and the floor through an abrasive paper, apply the new sealing ring.
Install the bowl again on top of the drain flange and press gently until the wax forms a seal, tighten each of the t-bolts.
Put the water supply pipe back and turn on the water supply, flush the toilet and check for leaks, if any.
The flush tank
Find out when does your tank leak and where the leak is coming from. Is it because of an old, cracked rubber washer or has your tank developed a crack? Cracks can be either hairline cracks, which can be fixed through Styrofoam insulation or special epoxy seals available at your hardware store, or they can be small cracks - in this case the tank needs to be replaced.
In case the leak is due to an old washer only, turn off the water supply and empty the flush tank.
Unfasten the bolts connecting the tank to the bowl and open up the water supply pipe. Remove the flush tank and look at the washer situated at the bottom of the tank where the water supply pipe to the bowl attaches.
Put the new washer and install the flush tank back on the bowl. Connect the water supply pipe back and turn on the water. Make sure that there are no leaks now, and the job is finished.
Water supply pipe to the flush tank
Turn off the water supply faucet for the tank, open the pipe from the faucet and the tank, discard it, pick up a new one with similar dimensions, connect it to the tank and the faucet, and turn on the faucet. Done!
Water leakage from flush tank into the bowl
Generally the most common problem, this is caused due to a faulty seal inside the flush tank known as the flapper. Get a new flapper from the hardware store. To install it, turn off the supply and discard the old flapper. Put the new one in place and then turn on the supply. Test the apparatus by flushing the toilet. Adjust the length of the chain if your toilet does not flush properly. Keep adjusting and flushing until you have a well-tuned flush tank.
Overflowing of the flush tank
This is perhaps the next most common problem after the flush tank leakage into the bowl. If you see your flush tank overflowing, you need a new fill valve. These are available in two styles - one with the arm and a big float ball and the second with the float ball installed on the fill valve itself. Any of them can be used as both are mutually replaceable.
To fix it, purchase a new fill valve, turn off the supply, and disconnect the water supply pipe.
Empty the flush tank, unfasten the fill valve and discard it.
Put in the new fill valve at the right height in the flush tank.
Now fasten the bolts. Connect water supply pipe and turn on the supply. Make sure that there are no leaks.