The taming process is where things get really cool. You'll need to knock out your desired dinosaur, then feed it food and potions to keep it alive. Once it's tamed, you can ride it, feed it, and even breed it with other dinosaurs to create new, awesome species.
There are so many satisfying moments in this game, like when you finally tame that one dinosaur you've been chasing for hours. Or, when you build a massive base and fill it with all your favorite creatures.
ARK Survival Evolved: How to Tame - Scalacube