This is actually a really hard question for a lot of reasons, notably because the physics that govern clouds on a small scale are quite complicated, also because there's lots of different types of clouds, but broadly speaking ... :
The droplets in the cloud start to coalesce with each other and get bigger and bigger until they fall out the sky.
This happens either because the cloud gets more and more water vapour added to it (for instance from strong updrafts which carry moisture up from the ground, this forms little fluffy white clouds of cumulus, or in extreme cases cumulonimbus, when thunderstorms happen).
It could also be because the water laden air is moving over higher ground and being forced upwards, this cools it and, like when you take a bottle of coke or similar out the fridge, water droplets form by condensation.
In the UK frontal systems predominate, in this case the same thing happens but the air is being forced up by cold air moving underneath warmer air, so the warmer air cools, the water vapour starts to condense and when the droplets reach the right size, they fall out the sky.
Warm air holds more vapour than cold air, which is why you can often tell it is about to rain when there is a sudden drop in temperature and a little blast of cold air.
I admit it. I'm a nerd...