Children of Ilúvatar
The Children of Ilúvatar are the two races of Elves and Men. Their existence, along with the Universe in which they were to live, was conceived in the Music of the Ainur, but the Ainur had no part in their making. Ilúvatar alone brought them into being, and they had fëar (spirits) of the same type as the Ainur, though far less powerful.
Unlike the Ainur, however, the Children were intended to exist as a union of flesh and spirit; their bodies, or hröar, are as integral to their being as their fëar. As such, they are also called Mirröanwi, or Incarnates.
While their status as embodied souls created by Ilùvatar makes them extraordinarily alike, each Kindred of the Children was also bestowed with different gifts from their maker. The Elves received great beauty, near immortality, skill with arts and crafts, and keen memories. Men, on the other hand, were given freedom to act outside the boundaries set by the Music of the Ainur, eventual supremacy over Middle-earth, and mortality.
The Dwarves, while also incarnates, are not typically called Children of Ilúvatar, for though they were given sapience and independent being by Eru, they were in fact created by Aulë. When Eru gave blessing to Aulë's work, he distinguished the Dwarves from Elves and Men by calling them "the children of my adoption" while referring to Elves and Men as "the children of my choice."
Parallels between Men and Elves
The history of Men and their races throughout the Ages, bore some similarities to that of the Elves, especially during the First Age.
Eventually some of them (Calaquendi and Númenorians) passed the Great Sea and reached a western place (Valinor and Númenor). They advanced and prospered near the Light of Valinor while their kin were left behind in Middle-earth.
There, some were corrupted by the Enemy which caused a Fall (Noldolante and Atalante). A part of them returned to Middle-earth and founded realms (Exile of the Noldor and Realms in Exile) becoming Kings among their "lesser" kin.