Most coloured diamonds are made synthetically because the variables necessary to form a naturally coloured diamond are extremely rare, thus why they are the most expensive kind of diamonds.

The most common way diamonds are made synthetically is HPTC – High Pressure High Temperature

Because the video (1:51) has no dialogue, here is what’s happening:

1. Pure carbon and catalyst metals, pressed into a graphite, are put into a small cylinder, and then placed in a growth chamber. 
2. The growth chamber is placed into a BARS apparatus consisting of a steel outer anvil and a tungsten carbide inner anvil.
3. For 3-5 days, the growth chamber will undergo extreme temperatures and pressures (1400 °C and ~880,000 pounds per square inch).
4. Once the ideal conditions are reached, the graphite dissolves into a molten metal solution, and the carbon atoms slowly build upon a crystal structure.
5. Once the capsule finishes cooling, cracking open the growth chamber reveals a rough diamond ready to be cleaned and cut.

Different types of metals added to the growth chamber will change its color. Eg. Boron will result in a blue diamond.