Basically it comes down to this - there are two forces acting on a parcel* of air in the atmosphere:
1. Gravity pulls downward.  The strength of the gravity force depends on the mass of the air inside the parcel.  This force is just the weight of the parcel
2. There is an upward pointing pressure difference force.  This force is caused by the air outside (surrounding) the parcel.  Pressure decreases with increasing altitude.  The pressure of the air at the bottom of a parcel pushing upward is slightly stronger than the pressure of the air at the top of the balloon that is pushing downward.  The overall effect is an upward pointing force.

When the air inside a parcel is exactly the same as the air outside, the two forces are equal in strength and cancel out.  The parcel is neutrally bouyant and doesn't rise or sink.

If you replace the air inside the balloon with warm low density air, it won't weigh as much.  The gravity force is weaker.  The upward pressure difference force doesn't change (because it is determined by the air outside the balloon which hasn't changed) and ends up stronger than the gravity force.  The balloon will rise.

Conversely if the air inside is cold high density air, it weighs more.  Gravity is stronger than the upward pressure difference force and the balloon sinks.

* the word parcel just means a small volume of air. 

We'll look at this in a slightly different way in class on Friday.  We'll find that when something is less dense than the gas or fluid surrounding it, the object will rise or float (wood floats in water because the wood is less dense than the water).  Objects that are denser than the surrounding fluid will sink ( a rock thrown into water will sink).

We did a short demonstration to show how density can determine whether an object or a parcel of air will rise or sink.  We used balloons filled with helium (see bottom of p. 54 in the photocopied Class Notes).  Helium is less dense than air even when the helium has the same temperature as the surrounding air.  A helium-filled balloon doesn't need to warmed up in order to rise.

We dunked the helium-filled balloon in some liquid nitrogen to cool it and to cause the density of the helium to increase.  When removed from the liquid nitrogen the balloon didn't rise, the gas inside was denser than the surrounding air (the purple and blue balloons in the figure above).  As the balloon warms and expands its density decreases.  The balloon at some point has the same density as the air around it (green above) and is neutrally bouyant.  Eventually the balloon becomes less dense that the surrounding air (yellow) and floats up to the ceiling.

Something like this happens in the atmosphere.