The movement of liquid molecules from the surface of the liquid to the gaseous phase above it is known as?

Molecules at the surface of a liquid with enough energy escape into the gas above it. This is evaporation. It happens below the boiling point and makes the liquid cooler.

Condensation is the opposite: gas becoming liquid. Liquefaction is turning a gas into a liquid by pressure or cooling. Brownian movement is the random movement of small particles in a fluid.