How do storms form?
Meeting of cold front and warm front
When two different fronts collide, rainstorms and snowstorms form. When a warm front moves in and meets a cold front, heavy nimbostratus clouds develop. When a cold front moves in and meets a warm front, comulonimbus clouds produce thunderstorms.
It is entirely dependent on what type of storm you are talking about.
As far as storms go, there are many processes involved. The only thing in common about all storms is there is a lifting mechanism and there is enough humidity.
Clouds form when moist air is cooled to the point where it is holding all the water vapor it can (100% relative humidity), which causes the water vapor to condense into water droplets or clouds. Clouds are necessary for storm formation.
The different ways that an air mass can be lifted (which is how it cools) include both warm and cold fronts, but also include orographic lift (hills and mountains), convergence (low pressure or trough), convection (unequal heating of the Earth's surface) and turbulent mixing (strong winds over rough terrain producing eddies that lift the air).
Dependent on the atmosphere stability at the point where the lift occurs will determine how much cloud forms, if a storm forms and what time of storm forms.
As condensation occurs, latent heat (heat stored in the water vapor) will be released causing the lifted parcel of air to cool at a slower rate. If that parcel of air then continuous to be lifted it may eventually be lifted to a point where it becomes warmer than the surrounding atmosphere. At this point (called the level of free convection) the parcel will rise on it's own due to buoyancy. If the parcel rises high enough it may produce a cumulonimbus cloud and a thunderstorm will occur.
In the case of a storm from a nimbostratus cloud, the air mass has to be very moist. This sort of cloud is usually seen at a warm front (retreating edge of a cold air mass). At this point warm air will move against the back edge of the cold air mass and rise up. Due to the slope of the frontal surface this type of lift can produce a very widespread cloud that can be up to around 20,000 ft thick. Certainly not thick enough to produce a thunderstorm, but thick enough to produce moderate to even heavy precipitation on rare occasions.
If you are interested in knowing how other types of storms form please ask a specific question about that type of storm.