For areas where the ground is heated by the Sun (especially the tropics), convection rainfall is common. The Sun heats the ground, which in turn heats the air just above the surface. This hot air rises, hits the dew point, forms clouds and then precipitates. This is why tropical areas often have heavy falls of rain in the afternoon.
Frontal rainfall is the result of two bodies of air, of different temperatures (and therefore pressures) meeting. Warmer air is less dense, it is forced over the cold (heavier) air mass. The warm air is transformed to cooler air as it rises and hits the dew point. Frontal rainfall can provide moderate to heavy falls. The formation of fronts can be as the result of global wind currents, localized events or geographic features (e.g. mountain chain).
Relief, or orographic rainfall, occurs when air has crossed a sea or ocean (and has become moist as a result) and then hits a mountain side. Warm, moist, air is forced up the mountain side, where it cools, condenses and precipitates. There is little rain on the far side of the mountain - this is called the 'rain shadow'.