According to the Met Office, the definition of precipitation is any form of water (liquid or solid) falling from the sky. This includes rain, sleet, snow, hail and drizzle, along with a few other less common precipitation types, such as ice pellets (graupel) and freezing rain. Precipitation is one of the mechanisms by which the atmosphere can directly interact with our lives on the surface. Although we often think of precipitation as a hindrance to our lives - getting wet when it rains and travel disruption caused by snow and freezing rain - falling precipitation ultimately sustains life on Earth and our existence. Precipitation is the primary mechanism for transporting water from the atmosphere to the surface and plays a fundamental role in the hydrological cycle. 
A Brief overview of Precipitation
Rain is perhaps the most common of all types of precipitation: it consists of liquid drops of water falling from clouds due to their own weight. When water droplets within the cloud become too heavy to be suspended by air currents and updrafts within it, they fall back down to the surface as rain. Raindrops are typically 1 - 4 mm in diamater, although in extreme examples they can reach up to 10mm - at this size, droplets tend to become unstable and break up. A droplet less than 0.5 mm in diameter is classed as drizzle - these droplets form from relatively thin stratus clouds and are  just heavy enough to fall out of the clouds. There are three main types of rain: frontal rain, orographic rain and convective rainfall. 
Snow - at temperatures well below freezing, ice crystals can grow in the atmosphere, as water vapour deposits onto a surface suitable for ice crystal growth - known as an ice condensation nuclei. These crystals can grow by sticking together (a process known as aggregation), and when an aggregate becomes heavy enough, it falls to the surface as a snowflake. 
Sleet  is a mixture of rain and snow. There is no agreed definition - in the USA, it is known as ice pellets or graupel - but in the UK it is known as snow which is in the process of melting as it reaches the surface. Surface air temperatures must be above freezing. 
Hail - unlike other forms of precipitation, hail can only form in convective clouds, such as cumulonimbus clouds. A hailstone grows in a region of the atmospere which is well below freezing, typically in the middle and upper parts of a thunderstorm. An ice crystal can grow by aggregation, in the same way that snowflakes form, or by riming - when supercooled liquid droplets make contact with the crystal and freeze instantly to it, gradually increasing its size. Hailstones grow so large because they are suspended by the updraft of the thunderstorm, and constantly grow as supercooled droplets and small ice crystals which are ascending in the updraft, make contact with them. Eventually, the hailstone will become too large to be suspended, and it will fall back to Earth. A hailstone must be larger than 10 mm in diameter to be classed as hail - otherwise it is graupel, but stones can grow up to 150mm in the strongest thunderstorms. 
Graupel (ice pellets) - these are essentially frozen water droplets. They form when snow initially falls from the clouds and then melts in a layer of air which is above freezing. However, crucially, the temperature close to the surface must be below freezing so that these liquid drops then have time to re-freeze into ice pellets before they reach the ground. They are uncommon in the UK, and typically bounce when they hit the ground.
Freezing rain forms in a similar environment to ice pellets - however, the below-freezing layer close to the surface is less cold. It is not cold enough to fully re-freeze the droplets: instead they become supercooled, so that when they touch anything at the surface - cars, trees, buildings etc. they instantly freeze. This forms a thin layer of ice on anything at the surface, and can cause treacherous driving conditions. 
Diamond dust - this is essentially just individual ice crystals that form close to the surface. In order for this to occur, temperatures need to be below -30°C, so this is most likely to occur only in very cold regions during winter. The individual crystals are small enough to remain suspended in the air, and the name comes from the sparkling effect when sunlight reflects off of the ice crystals. 
Close-up photo of snowflakes. Taken on 16th November 2014, in Norman, Oklahoma. 
A large hailstone dropped by a tornadic supercell in northern Texas, on 22nd April 2015.
Glazing on tree branches caused by freezing rain. 
Diamond dust can generate spectacular halo phenomenon as seen here, at Krkonose in the Czech Republic.