They're made of carbon, just like natural diamonds. The most common process involves large presses that can weigh hundreds of tons to put carbon under tremendous pressure and high temperatures, essentially mimicking the process by which they naturally form. There's also another method called chemical vapor deposition, which creates a carbon plasma over a substrate onto which the carbon atoms deposit to form a diamond.
It's made the same way natural diamonds are, and it's made of the same thing - carbon atoms!
All you need to make diamonds is high pressure and/or high temperature, according to [carbon's phase diagram](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Carbon-phase-diagramp.svg/467px-Carbon-phase-diagramp.svg.png
). The phase diagram shows which state of matter a compund exists in at a specific pressure/temperature.
Natural diamonds are formed several hundred kilometers below the Earth's surface, where the pressure and temperature is high enough for diamond to be the stable structure. Most synthetic diamonds are made the same way; you have carbon atoms, and you increase the heat and pressure to the point where they form a diamond structure instead of graphite.