In order to reproduce dinosaurs in ARK: Survival Evolved you need to tame at least two of the same species so you are able to breed them and incubate their eggs. To make your tamed dinosaurs willing to mate you must prevent them from following you by enabling Wandering mode. This is done by highlighting the dinosaur and then tapping the dinosaur skull icon in the top-left corner of the screen. Then in the dinosaurs menu tap the 'Enable Wandering' checkbox. Repeat the process with the second dinosaur. Once you have done this you need to place both dinosaurs inside a small pen and wait. The amount of time needed before you get an egg will vary depending on the dinosaurs you are trying to breed and can be anything from 15 minutes to 3 hours. If the dinosaurs are mating dark red heart icons will appear above their heads and the bar beneath their names will change to 'Mating Progress' and a timer. When the timer reaches 100% you will get an egg. Make sure before you start incubation that the egg is labeled as a fertilized egg otherwise it will not hatch.
The fertilized egg then needs to be heated for a period of time in order for it to hatch, this can be achieved by placing it next to a heat source that has the required temperature. The hatching process will begin when the text on the egg changes to 'Egg Incubation 100%'. Highlight the baby dinosaure once it has hatched and select the dinosaur skull icon, then choose 'Claim' to immediately tame it. After this you need to transfer the baby dinosaur to the new baby's inventory which will begin the maturation process.