Everything Ehsan said is correct. "Flat" is British, and "apartment" is American. They both mean the same thing.
"Floor" means which level you live on. Example: "my flat / apartment is on the second floor".
However, I would just add that "flat" and "floor" can also have other meanings. (Apartment has no other meaning apart from the one already stated).
"Flat" can be an adjective. It means smooth, with no bumps or obstacles. Example: "this road is flat. There are no hills. It is easy for me to ride on this road on my bicycle."
"Floor" can also mean "the place where we stand inside a house". Example: "the floor in my house is covered with carpet."