There are a number of reasons for separating a tank from the toilet bowl.
There are usually two or three bolts holding a tank and bowl together. Review the Toiletology 101 “anatomy” pages to get an idea of how the two pieces fit together. Turn the water supply off, then completely drain the tank and sponge up any water in the bottom. Disconnect the water supply line.
Use penetrating lubricant (LPS-1 or WD-40) on the nuts under the tank. Give the penetrating lubricant time to work…overnight if possible. Holding a large screwdriver on the head of the bolt inside the tank, turn the nut under the tank with an adjustable wrench. If the nuts refuse to come loose, you may have to resort to sawing through them with a hack saw. It’s a tedious job, but it can be done. Once separated, you should find a cone-shaped gasket and a tank cushion gasket. You will have gone to a lot of work to get this job done. I suggest that you replace all the washers, bolts and gaskets that you can with new ones; you must at the very least replace the washers and cone-shaped gasket. Now is the time to install a new flush valve if you need one.
You’ll need a tool called a “spud wrench” to undo the nut that holds the flush valve in place. When you go to buy a new flush valve make sure the overflow pipe is the same height as the original one. This is important. If the overflow pipe is too high, water will leak out of the hole for the handle and flood your house if the ballcock fails.