In the beginning of the game you don't have the tyke, you get it after you talk to some of the villagers that live there, they tell you to go to a three and throw out the egg that's laying in there.
Ones you've done that you will need to put in a place that's warm enough to make it hatch and after a while you will see the tyke jumping out off the egg.
As you will see tyke is a chicken and it seems that he is very smart, you can compare his intelligence with an ape (roughly)
The meaning of tyke is to take good care of him and to teach him so that Eve will see how good of a parent your creature can be.
You will have to teach him a couple of miracles, teach him to build and some other stuff aswell.
Although your creature's tyke is a fast learner it can be hard to teach him some of the simple stuff like letting him sleep at night. If your creature didn't taught him that properly one of the things he can do is bugging your creature all night and not letting him sleep and thus letting him grow a lot slower, or tyke decides he doesn't want to eat and passes out every 2 minutes, these are some of the points you really have to keep a close eye on when training him, especially in the beginning.
When your creature has taught tyke the necessairy things he will have completed a trial, when he has completed all trials he will get a son together with Eve.
This son will have exactly the same stats your old tyke had, he will just have another skin and he might have changed his fighting style.
The tyke changes in an animal that is simular to your own creature, it will make it look like a minitature version of himself.