Hi hollywing, welcome to Rescreatu! :D
When you hatch an egg, it will either fail or turn into a Creatu. If it fails, you won't receive a pet, and you'll have to try again with a new egg. If your hatch is successful, you will be brought to a naming page where you can name your pet and view its picture, species, color, gender, and birth date. If you exit this page, you can return to it later by going to your hatchery and clicking the link at the top of the page that says something like, "You have 1 Creatu awaiting names." Once you give it a name, the Creatu will be yours to take care of and will be transferred to your profile! Just make sure to give it a name within 15 days, or your Creatu will get sick of sitting in the hatchery and will wander off forever.
Either way, once you hatch the egg, the egg itself is gone. It is a one-time-use item, meaning that you can only try to hatch it once before it disappears.
If you have any other questions, I would be happy to help! Feel free to Rmail me if you need to, or I will be here on this thread.
I hope you're enjoying yourself so far, and I'll see you around. :)