Clouds can be formed in many different ways. Before we go through them, we need to establish some basic facts.
Clouds are made up of liquid water, or solid ice. Clouds are not water vapour; vapour is a gas and is invisible. Once more, because this is important: clouds are liquid water, not water vapour.
Water droplets in clouds are extremely small. Clouds are visible because light is scattered in these droplets. The water droplets can stay suspended in the sky due to friction and air currents.
In order for water vapour to condense into liquid water droplets, to form clouds, condensation nuclei must be present. This can be tiny particles of dust or other impurities around which water droplets can form. This is where the idea of human cloud generation comes form - by adding condensation nuclei to the atmosphere, clouds are more likely to form.
A mass of air can contain a certain amount of water vapour (=gas). The maximum amount depends on several factors, of which the most important one is temperature. Cold air can contain less water vapour than warm air. If more water vapour is present than the air can contain, the water has to condense (the opposite of evaporate), to form water droplets - clouds.
So, how are clouds formed then?
By adding moisture
Air can only contain a certain amount of water vapour. If we try to add more water vapour to the air, it "won't fit". As a result, the water vapour will condense, forming visible water droplets - clouds. This can happen over a surface of water during a warm day, when surface water evaporates. The air won't be able to absorb the evaporated water, so instead it has to become liquid.
Cooling from below (advection)
Warm air can contain more water vapour than cold air. When a warm, moist airmass moves over a cold surface, the temperature of the air will drop. When reaching a certain temperature, the air becomes saturated - it contains exactly as much water vapour as it possible can. We call this temperature the dew point. If the air is cooled further, some of the water vapour in the air has to condense into water droplets. Imagine squeezing a saturated sponge - the water has to come out.
Adiabatic cooling (convection)
On a sunny day, the sun will heat up the surface of the earth. Because the surface is not uniform, it will heat up at different rates. A sandy field will get much warmer than a wet area, for example. The air directly above such hot areas will also be heated up, by radiation from the surface. Hot air has a lower density than cold air, so at some point it will start to rise ("heat rises"). As this warm bubble of air rises, it will start to expand. This is because pressure drops with altitude. As the air expands, it will cool off. Now we're back at the basic principle that cold air can hold less water vapour than hot air. As our bubble of air rises, and cools down, at some point, it will reach the dew point, and a cloud will form.
The adiabatic lapse rate is the rate at which a bubble of rising air will cool off with altitude. Rising air will cool with approximately 3 degrees C per 1,000 ft of altitude gain (dry adiabatic lapse rate - DALR), simply because it expands as a result of the lower pressure. When air starts to condense, and a cloud forms, this process releases latent heat. Simply put, it took energy to make the water evaporate, and this energy is released again when the water condenses. This means that once our bubble of rising air reaches the dew point, and a cloud starts to form, it will actually cool off slower. The saturated adiabatic lapse rate (SALR) is approximately 1.5 degrees C per 1,000 ft.
Now, for how long will our bubble of air keep rising? It will keep rising as long as the air surrounding it is cooler than itself. The general temperature of the atmosphere at a given location is known as the enviromental lapse rate - ELR or ELT. As long as the DALR is less than the ELR up to the dew point, rising air will form clouds.
A warm front is when a mass of warm air moves forwards, displacing a mass of colder air. Since warm air has a lower density than cold air, when the two air masses meet, the warm air will be forced upwards, above the cold air. This causes the air to expand, and cool. As this happens, the air might reach the dew point, and clouds will be formed.
Orographic lift is quite similar to frontal lift. Instead of being forced up by cold air below, an air mass is forced upwards by terrain (i.e. wind blowing towards a mountain). Then it's the same old story - the air rises, expands, cools and water vapour condenses.
Turbulence and convergence
Mechanical turbulence, for example due to rough terrain, can cause vertical movements in the atmosphere. If these are large enough, air can be forced upwards, which, as we now know, will form a cloud.
Convergence is when two air masses move towards each other, creating a vertical stream of air in the middle. This column of air moves upwards, and clouds are formed.
These are the basics of cloud formation. As you can see, they all depend on the same basic principles that I covered in the beginning of this answer.