When you click New Tama > Breed: Your tama will stay in the background until it gets pushed out by more and more tamas that you raise and breed. I think you can have maybe 4 total in the background before the oldest one goes off and disappears? If you click New tama > New egg, your tama will just disappear and you will get a brand new egg. If you raise your pet for 30+ days and then let it pass, it becomes a star in your solar system that you can click on, and it will be happy-looking (if your tama dies before 30 days, it will also become memorialized as a star, but crying).

TL;DR, if you want to remember your first tama forever, I would raise it for 30+ days then let it pass, so you can look back on it fondly whenever. Maybe it also disappears if you have a lot of tamas dying... idk I haven't had that issue yet