To establish communication and initiate interaction with a time-shared computer or network — often used with to

Entrance to a computer system, condition of being connected to a computer system
Or log in; In computing, the process by which a user identifies himself or herself to a multiuser computer and enters the system. Logging on usually requires the user to enter a password before access is allowed.
The process of identifying oneself to a computer after connecting to it over a communications line. Also called: login. log on.