He belongs to the species Tatsumi, similar to dragons (the Japanese dragons are quite different from Western dragons, and they are recognizable by the characteristic shape of the horns). His name comes from Tamago, which means "egg". It was chosen by Nyanko-sensei. His name is not contained in the Yuujinchou.
Tatsumi Youkais do not have a definite form when they are still young. They take upon the same form as the first thing they see, believing that person/youkai/animal is its mother. Because Natsume was the first thing "Tama" saw, it took on the form of a human. The only defining feature all young Tatsumi Youkai have are the horns.