The air next to the ground cools during the night.  Sometimes it cools enough to reach the dew point.  Water vapor condenses onto objects on the ground and you find everything covered with dew (or frost) the next morning.  When this happens in the air up above the ground you might think that water vapor would simply condense and form little droplets.  This is not the case; we will find that small particles in the air called condensation play an essential role in cloud (and fog) formation.

 it is much easier for water vapor to condense onto small particles called condensation nuclei rather than just condensing and forming small droplets of pure water

When the air is saturated with water vapor (the relative humidity is 100%) the rates of evaporation and condensation above a flat surface of water will be equal.

There's no real reason for picking three arrows each of evaporation and condensation, the important point is that they are equal when the RH is 100%.

It's hard for water vapor to condense and form a small droplet of water because small droplets evaporate at a very high rate.  This is known as the curvature effect and is illustrated below.

The surface of the smallest droplet above at left has the most curvature and the highest rate of evaporation (6 arrows).  If a small droplet like this were to form, it wouldn't stay around very long.  With it's high rate of evaporation it would quickly evaporate away and disappear.

The middle droplet is larger and would stick around a little longer because it does not evaporate as quickly.  But it too would eventually disappear.

The drop on the right is large enough that curvature no longer has an effect.  This drop has an evaporation rate (3 arrows) that is the same as would be found over a flat surface of water.  A droplet like this could survive, but the question is how could it get this big without going through the smaller sizes with their high rates of evaporation.
A droplet must somehow reach a critical size before it will be in equilibrium with its surroundings.

Particles in the air, cloud condensation nuclei (CCN), make it much easier for cloud droplets to form.  The figure below explains why.

By condensing onto a particle, the water droplet starts out large enough and with an evaporation rate low enough that it is in equilibrium with the moist surroundings (equal rates of condensation and evaporation).

There are always lots of CCN (cloud condensation nuclei in the air) so this isn't an impediment to cloud formation.  The following information is from p. 91 in the ClassNotes.

Note that condensation onto certain kinds of condensation nuclei and growth of cloud droplets can begin even when the relative humidity is below 100%.   These are called hygroscopic nuclei.  Salt is an example; small particles of salt mostly come from evaporating drops of ocean water.

To understand how this can occur we first need to learn about the solute effect

 solution droplet pure water droplet

Water vapor condensing onto the particle in the left figure dissolves the particle.  The resulting solution evaporates at a lower rate (2 arrows of evaporation).  A droplet of pure water of about the same size would evaporate at a higher rate (4 arrows in the figure at right).  Note the rates of condensation are equal in both figures above.  This is determined by the amount of moisture in the air surrounding each droplet.  We assume the same moist (the RH is 100%) air surrounds both droplets and the rates of condensation are equal.

The next figure compares solution droplets that form when the RH is 100% (left figure) and when the RH is less than 100%.

 the droplet is able to grow the droplet is in equilibrium with its surroundings even when the RH is less than 100%

The solution droplet will grow in the RH=100% environment at left.  You can tell the RH is less than 100% in the figure at right because there are now only 2 arrows of evaporation.  But because the solution droplet only has 2 arrows of evaporation it can form and be in equilibrium in this environment.

Don't worry too much about all the details.  The key point is that particles help clouds to form.