I think the easiest way to understand this is to examine why a helium balloon goes up. At school they teach you it's because it's "lighter than air", the force up is due to the difference in masses displaced. If the mass of air displaced by the balloon is MORE than the balloon, the difference is the force upwards.
Yea, that's an easy way of looking at it, but it doesn't tell you where the force is coming from. The force up is actually due to the difference in air pressure on the top and bottom of the balloon which is equal to density*g*h. The very slight difference in atmospheric pressure between the top and bottom of the balloon causes a slight upward force. If that upward force is greater than the balloon's weight (density*Volume) then the balloon will be forced upwards by the air pressure.
Now imagine helium coming out of a valve into atmosphere. The helium atoms bounce off the air molecules like a crowd of people pushing through a doorway into a larger crowd of people. The helium molecules can't disperse into the air instantaneously, so they tend to bunch up as they come out, just as you'd have a hard time pushing your way through a crowd.
So this volume of helium, with perhaps a small amount of air that's gotten into it is less dense than the surrounding air which has only a small amount of dispersed helium in it. Before the helium can disperse in the air, it actually is forced upwards because the air pressure underneath is higher than the pressure above and the weight of the helium is less than this difference, just like the balloon.
The same can be said for liquids. The reason a balloon filled with less dense liquid will float in a higher density liquid is because of the difference in pressure between the top and bottom which results in an upward force.
And the same can be said of a liquid that has no balloon around it. In fact, the collisions are much more numerous and there is a much slower diffusion rate of liquid in liquid. So the reason a lighter liquid can be made to float is because of the difference in pressure. As the molecules start to mix by diffusion, the barrier will eventually go away.
The heavier molecules will displace the lighter, forcing them to the top. Thus warm air or water will tend to rise to the surface.
This isn't exactly true. The only reason warm air or water tends to rise is because of the above, there's a rate at which the warm and cold air can mix by diffusion or convective currents, and until they get mixed, the warm air or water rises because it is being pushed upwards by a difference in pressure of the colder surrounding fluid. If for example, you have an insulated container, the contents will eventually come to some thermal equilibrium, and there will be no significant density gradiant other than that created by gravity. If we put helium in a room with nitrogen, there would similarly be very little separation between the two. The two gasses would eventually diffuse into each other and be very well mixed. The small difference in concentration between the top and bottom of the room has to do with the velocity of the molecules/atoms of the gas. It's a calculable difference, but relativly slight.
