I have two drains in the floor of my basement. When there is very heavy rain the drains back up and flood the basement. I am wondering why this is happening and what can be done to correct the problem. Can the drains just be plugged up? Or will this just cause the water to come up somewhere else, like in the toilet in basement?

You can insert a stand pipe if you can get a water tight seal at the floor. You could also break up the floor and put in a floor drain with a check valve. Flood guard works well too. Flood guards for floor drains - check valve to prevent flooding from FAMOUS PLUMBING SUPPLY

You could plug the floor drains, but I think you are right about it just causing problems elseware. Sounds like your sewer system is tied in with your storm drain system. In some areas where they have a common sewer/storm drain, there will be a valve in the basement floor that you can shut off in case of heavy rain.