Achievements are things that the game rewards you for doing different tasks. These tasks vary from leveling a mining laser to level three or winning a certain amount of PVP matches, et cetera, et cetera.
the rewards you obtain from achievements will vary. Common rewards award you with things like starbux, or normal currency in-game (gas or minerals). You may also receive a pin from an achievement, this pin may be equipped by your character. It appears whilst your character is in a dialogue bubble and can be seen externally in game.