All colors represent different jobs and service roles in a ship. The reboot series adapted colors from the original series.
Here is the list of what job each color represents.
The command division wore gold shirts. This includes Captain Kirk, Lieutenant Sulu, and Pavel Chekov.
The science and medical staff wore blue shirts. This includes Lieutenant Commander McCoy and Commander Spock.
The engineering and communications division wore red shirts. This included Chief Engineer Scotty and Communications Officer Uhura.
The security division also wore red shirts. You might know these people as the careless supporting characters that are immediately killed whenever the crew is confronted by a new enemy. We told you to avoid them.
When Kirk asks Chekov to put on a red shirt, he made Chekov his new chief of Engineering division.
KIRK: Mister Chekov. You've been shadowing Mister Scott. You are familiar with the engineering systems of this ship?
CHEKOV: Affirmative, sir.
KIRK: Good. You're my new Chief. Go put on a red shirt.
As indicated in the list of uniform colors, Chekov was asked to put on a red shirt.