At a certain height, air cools enough for any water vapour to condense into droplets and form visible clouds. The droplets are liquid water, and therefore denser than air, but they are tiny, so they have a low terminal velocity and fall very slowly. A typical cloud only has about 0.5g of water per cubic metre in it, and if the droplets are small enough, they will be kept aloft by the thermals in the cloud as warm air rises from below. Once the droplets have fused together and grown large enough, gravity dominates over buoyancy and they fall as rain.
As for why water that evaporates rises up into the sky in the first place, air is mostly nitrogen (N2) and oxygen (O2), with an average density of 1.225kg/m3. A water vapour molecule is much lighter with just one oxygen atom and two hydrogen atoms (H), so its density (at standard temperature and pressure) is only 0.804kg/m3.