Convective rain is produced by convective cloud. Convective cloud is formed in vertical motions that result from instability of the atmosphere. One way that the atmosphere can become unstable is by heating from the sun. The ground warms up, causing moisture in the ground to evaporate and rise, and the hot ground also heats the air above it. As the water vapour rises, it cools and condenses into clouds and eventually rain.
When you heat the air from below like this, much like in a boiling kettle, you tend to get "bubbles" of rising air, known as updraughts. These are much smaller than the large-scale lifting of air that occurs at fronts and over mountain ranges. This tends to give us smaller areas of rain, with clear spells in between, commonly referred to in the UK as "sunshine and showers".
This type of rainfall is most common in the south and east of the UK, where it is typically warmer. This area is also prone to very heavy showers and thunderstorms, this is because the warmer air can hold more water.
Sometimes, you can get all three types of rain at once, and this can lead to severe flooding.