Short answer is VERY heavy. To understand, lets think about what clouds are made of; either water droplets or ice crystals. Both liquid water and solid ice have a much higher density than air. OK, so why do they stay up?

For the same reason that dust, pollen, and the carbon particles from smoke stay up. They are very small. That does not make them less dense. (Small rocks don't float, in spite of what Monty Python says.) It does mean they have very little mass, so they are easily moved by air currents and even the brownian motion of atoms in the air around them.

OK, so how heavy? Think about what comes out of a cloud? Rain. Sometimes a LOT of rain. Imagine that a cloud drops one inch of rain on a one square mile area. That is 2,291,520 cubic feet of water. At 62.4 pounds per cubic foot, that is 142,990,848 pounds of water that came from the cloud, and there is still a lot of cloud left after the rain.

If clouds are that heavy, what would happen if one fell on you? Fog.