The first time I was in college, you'd talk to your advisor to figure out your schedule, then either call in to the phone system during registration hours, or go to an area set up with special computers and assistants, or, if all else failed, go talk to someone in the registrar's office.
More recently, it was all done online - you could work up your schedule in advance and were allowed to register after a certain time, and would log into an online system to do so. If you had any problems with that, you could call the registrar's office and usually it was a setting on your user account that they would fix ("oh, the payment for that library fine didn't get logged right, let me fix that for you..."), or they'd explain why the problem couldn't be fixed by them ("you need prerequisite XXX or instructor permission to enroll - just go to the department office, pick up form XYZ, have the instructor sign it, and bring it in"). If all else failed, your advisor could usually push through your class request. Only under the most dire of circumstances (ie, user account totally hosed through no fault of your own) could you just take your schedule in to an office and have them register you for all your classes. Plenty of folks available to help if you got an error or hit a snag, but you still had to largely do it yourself.