How you do it, and what extra factors there are (such as time limits, affection bars, gifts, and raising your own abilities) vary from game to game and have become more and more complex over time.
These themes are in a lot of today's games as well!
If you're looking to play a classic game some time to expand your dating simulator knowledge, True Love is still a fun play through.
True Love shows just where dating sims came from and where they are going.
It has most of the elements found in today's games, such as being able to improve your personal statistics to help your chances with the girls and having a time limit system in place forcing you to carefully manage and plan each of your moves.
From famous Japanese titles like Amnesia to Korea's incredibly popular Mystic Messenger, and free fan-made games like Re: Alistair - dating simulators are here to stay!