Bittergems are a rare resource in Deep Rock Galactic, and you need to find a total of 10 Bittergems to be able to unlock the trophy / achievement called "Jeweler". The Bittergems themselves are only awarding you with between 1000 and 1500 Credits, so they are not really required for anything, but they are a nice bonus whenever you find one.
The problem with Bittergems is that a lot of players play Deep Rock Galactic for hours and hours without ever finding one, and the reason for this is that Bittergems are very difficult to find unless you know what you are looking for.
To find and collect Bittergems, you need to look for small pink crystals that poke out of the rock (see screenshot below). These indicate that you should mine there to find a Bittergem. Once you find it, transport it back to the MULE to collect it.
Keep in mind that the Bittergems are extremely rare, and will only spawn in about 4.76% of all missions. They can spawn on any mission at any planetary region, but since they are extremely rare you will have to play a lot of missions before you will be able to find one.
On average, you can expect to play 210 missions before you have found all the 10 Bittergems that are required for the "Jeweler" trophy / achievement. And this is only if you search the entire map. In reality, you can expect to spend much longer, because you might not even find it in the mission if it has spawned.
When you mine the Bittergem, you must carry it back and deposit it to the MULE. While carrying it you cannot spring or use your weapons, and you will run 25% slower than normal.