After 24 hrs of evolving, you will be able to connect and mate with another Tama that is of the opposite gender and also 24hrs into its adult form.
After 72hrs or so, the Matchmaker, or Mrs. Busybody will come and offer your Tama a mate, so s/he can have babies.
If you ignore the matchmaker long enough your Tama will become an old timer. Then it will most likely die of old age as the matchmaker will not visit any more. However you can connect and mate with another old timer of the opposite gender.