The Children of Ilúvatar were Elves and Men, the two races created by Ilúvatar, the supreme being of created universe. Dwarves were not included in this distinction, as Aulë in fact created them. The story of their creation was told in the Quenta Silmarillion, and explains how the Elves and Men were created by Eru Ilúvatar directly, without delegation to the Ainur.
Elves were also named the Quendi ("those who speak", in Quenya) and the Minnónar ("Firstborn"). They were further divided into many groupings, often distinguished by their respective linguistic branches.
Men were also named the Apanónar ("Those born after" or "Afterborn"), Fírimar ("Mortals"), Engwar ("the Sickly"), and the Followers. These somewhat derogatory names were given to Men by the Elves, who were the primary historians of the early events of Middle-earth.
The reason for this naming was simple. Elves were the first of the Children of Ilúvatar to appear in Middle-earth, whereas Men were not to follow until the beginning of the First Age, with the creation of the Sun and Moon, many thousands of years later.
Animals and plants were probably fashioned by Ainur after themes set out by Eru in the Music of the Ainur. In cases where animals exhibit sapience, as in the case of Huan or the Eagles, this is questionable they may actually be Maiar instead.
Races of the Creatures of Arda