The word cover means to put something over or on top of something else. This can be for warmth or protection.
EX: Will you cover that pot? I want it to stay warm while I spread the vegetables on the platter and set the table.
EX: You will need to cover your head before going outside in this freezing weather. It is important to keep yourself from getting sick this winter.
The common phrase take cover means to find safety or shelter during an emergency or situation.
EX: If you are in a hurricane, it is recommended that you take cover in a doorway or bathtub to avoid flying or falling debris.
EX: The freezing rain was coming down so hard that we had to take cover in a roadside restaurant.
The word conceal means to put something on or over to keep out of sight or protect. In this case, it is a synonym for cover.
EX: You need to conceal that pile in the backyard. The elements will wreak havoc on the wood, and we will not be able to use it in the fireplace.
EX: You should conceal those dark circles under your eyes. They reveal how little sleep you are getting and how hard you are working.
The word conceal can also mean to keep something secret. In this case, it is a synonym for hide.
EX: In many mystery novels there is a character who is concealing their true identity. It keeps the tension going and adds to the entertainment when the person is revealed.
EX: I'm trying to conceal how much money I have in the bank. I feel like stockbrokers are always trying to approach me as a potential client.