Hot air rises because when you heat air (or any other gas for that matter), it expands. When the air expands, it becomes less dense than the air around it. The less dense hot air then floats in the more dense cold air much like wood floats on water because wood is less dense than water. This floating effect in a less dense medium is called a buoyant force or a displacement force.

We can model what I just said with a little physics. Consider the air to be an ideal gas (this is a good approximation which neglects the interaction of air molecules with each other).

The pressure P, volume V, amount of gas N, and temperature T can all be related by the ideal gas equation as
P*V=N*R*T
where R is the ideal gas constant (8.314 in units of J/(mol*K)).

Say you start with an amount of air N at pressure P, volume V₁, and Temperature T₁ and heat it in a balloon (we will neglect the mass and elastic properties of the balloon for simplicity) to temperature T₂.

The ideal gas equation can be rewritten as
P*V₁/(N*T₁) =R=P*V₂/(N*T₂)
which with a little algebra can be solved to give
V₂=V₁*T₂/T₁ .

From this we see that if you heat a gas such that
T₂>T₁
you get
V₂>V₁
meaning that the gas expands. Now since you have the same amount of gas, the mass does not change so the density before heating is
d₁=m/V₁
and the density after heating is
d₂=m/V₂.

Now the force of gravity on the hot air (F_g) is given by the mass of the hot air times the gravitational constant g (9.8 m/s²)
(F_g=d₂*V₂*g).
Here the mass of the hot air is given by the density of the hot air times the volume of the hot air. The buoyant force on the balloon is given by the mass of the air displaced times the gravitational constant (F_b=d₁*V₂*g).
Here the mass of the air displace is given by the density of the surrounding cool air times the volume of the balloon (hot air). Now the buoyant force is up and the gravitational force is down so the net force is given by
F=F_b- F_g=d₁*V₂*g- d₂*V₂*g=m*g* (V₂/V₁- V₂/V₂)
which can be rewritten using the relation from the ideal gas law (V₂=V₁*T₂/T ₁) to give:
F=m*g*(T₂/T₁-1).

This equation shows that as you heat the air in the balloon T₂>T₁ (i.e. T₂/T₁>1) you get a positive force (F>0) which means that the force is up and the balloon rises.

In summary, hot air rises because it is less dense than the surrounding air.

Hot air pushes out on the air around it, and thus becomes less dense. Less dense air rises because the denser, cooler air sinks because it is heavier and gravity pulls on it more strongly.

Most movement of particles (such as air) happens due to diffusion. If you put sugar or salt into water, it will eventually dissolve into the whole volume of water. In this example, it's because there is originally a different concentration at one point than another (that is, there's a lot of sugar sitting on the bottom of the cup, and none at the top). This creates a gradient, a difference in concentration in this case, which can do work (like move particles!).

Hot air rising happens for similar reasons. Hot air is less dense than cold air. When the air gets hot, its density decreases, and now there's more air molecules in the colder zone than in the hot zone. There are more complicated things happening, but for this simple reason alone, we might expect hot air to rise for the same reason that objects less dense than water will float in water.