How does rain form?
There are two primary methods by which rain is produced: the warm rain process and the cold rain process. In a cloud where the temperature remains above freezing throughout its entire depth, then rain will form through the warm rain process. However, in a cloud which is partially below freezing, then it is likely that rain will occur as a result of the cold rain process. As a result, the warm rain process is the predominant precipitation production process in the tropics, whereas it is the cold rain process which is most important in the mid-latitude regions. 
Warm Rain Process
In order for a droplet to grow in the atmosphere, it requires the presence of a cloud condensation nuclei (CCN). These can be bits of pollen in the atmosphere, or other aerosols such as fine volcanic ash, or soot from human factories. The largest natural source of CCN is from sea salt, which is released into the atmosphere from the formation of sea spray as waves break in the ocean. A liquid droplet cannot form on its own; it requires the presence of a surface in order for the water molecules to bind to. A liquid drop can also only form in an environment where the air is supersaturated - that means an environment where the relative humidity is greater than 100%. It turns out that due to the surface tension of a small drop of water, the air must actually be slightly supersaturated in order for water vapour to condense to form cloud droplets. 
Ultimately, some of the cloud droplets will become large enough just by the condensation of water vapour, to slowly fall to Earth. As they do so, they collide with other cloud droplets and coalesce together. This new droplet becomes even larger and heavier, and so falls to Earth faster, speeding up the growth process. This process of growth is known simply as collision coalescence.  The droplets can become too large by this growth method - they become unstable, and can break up when falling or when they collide, producing lots of smaller droplets that can sustain the growth of other droplets. 
Schematic depecting the collision-coalescence process. The large droplet falls faster than the smaller cloud droplets, and so collects them and grows. 
Cold Rain Process
It was soon noted after the acceptance of the collision-coalescence rain process, that this method could not work in the mid and high-latitudes, where a large portion of the atmosphere remains well below freezing, even in the summer. Therefore, clouds cannot consist entirely of liquid droplets, and so the formation of rain must start off its life in frozen form somehow. 
It is much more difficult for ice crystals to form in the atmosphere: like liquid drops, they to require a surface to initiate growth - called an ice condensation nucleus (ICN). However, these typically need to be in a hexagonal shape for ice crystals to grow on them, making ICN rather rare. Any liquid droplets will remain in a supercooled state until the temperature drops below around -40°C, at which point they will start freezing spontaneously. The majority of "cold" clouds therefore consist of a mixture of ice crystals and supercooled liquid droplets. The supercooled droplets vastly outweigh the number of ice crystals however. 
Ice crystals require a lower saturation vapour pressure than liquid water droplets. This means that ice crystals require less water vapour in the atmosphere than liquid droplets to grow by deposition. In most clouds, the environment is subsaturated with respect to liquid dops, but supersaturated with respect to ice crystals. This results with the liquid droplets evaporating, while the few ice crystals grow at their expense, until all of the droplets have evaporated and only ice crystals remain. These can grow large enough that they begin to fall out of the cloud. They may continue to grow by aggregation with other crystals, or by riming - where small supercooled droplets freeze directly onto the ice crystal surface. 
These ice crystals can aggregate together to form snowflakes, and can reach the surface as snow if the atmosphere remains below freezing. However, they can also melt to form rain at the surface. This is the method by which almost all of the rain in the UK is produced. It is called the Wegener–Bergeron–Findeisen process. 
Schematic showing the cold rain process. The ice crystals grow at the expense of the supercooled liquid droplets, until there are none left.