[RedFusion]

Register

[rej-uh-ster]

• noun
  • 1. a book in which records of acts, events, names, etc., are kept.
  • 2. a list or record of such acts, events, etc.
  • 3. an entry in such a book, record, or list.
  • 4. an official document issued to a merchant ship as evidence of its nationality.
  • 5. registration or registry.
  • 6. a mechanical device by which certain data are automatically recorded.
  • 7. cash register.

• verb (used with object)
  • 15. to enter or cause to be entered formally in a register.
  • 16. to cause (mail) to be recorded upon delivery to a post office for safeguarding against loss, theft, damage, etc., during transmission.
  • 17. to enroll (a student, voter, etc.) in a school or course of study, on the voting rolls, etc.
  • 18. to indicate by a record, as instruments do: The thermometer registered 102 degrees today.
  • 19. to indicate or show, as on a scale.
  • 20. Printing. to adjust so as to secure exact correspondence; cause to be in register.
  • 21. Military. to adjust (fire) on a known point.

• verb (used without object)
  • 24. to enter one's name or cause it to be entered in a register; enroll: to register at a motel.
  • 25. to apply for and obtain inclusion of one's name on the list of voters.
  • 26. to enroll in a school or course of study: I've registered for three English classes.
  • 27. Printing. to be in register.
  • 28. to show: A broad smile registered on his face.
  • 29. to have some effect; make some impression: My plea didn't register on him at all.

Source: http://www.dictionary.com/browse/register

tl;dr: It usually means to save your username and player data (such as kills, deaths, score, inventory, vehicles, configuration) for later use.
Usually this is done with a password which restricts other users from using your name and accessing your data.
And in some cases you have to provide your e-mail adress, which your new password will be sent to incase your account gets hi-jacked or you forget your password.