Dwarves Awakened (4615 F.A.) - The Dwarves were actually created before the Elves.  But not by Ilúvatar.  Aulë had grown impatient waiting for the coming of the Firstborn and he desired for the Earth to be filled with creatures to whom he could teach his ways.

 

"It is told that in their beginning the Dwarves were made by Aulë in the darkness of Middle-earth, for so greatly did Aulë desire the coming of the Children, to have learners to whom he could teach his lore and and his crafts, that he was unwilling to await the fulfillment of the designs of Ilúvatar.  And Aulë made the Dwarves even as they still are..." ~The Silmarillion, chap. 2 (Of Aulë and Yavanna)

 

But when Ilúvatar discovered what Aulë had done, he was displeased by this action.

 

"Ilúvatar spoke to him; and Aulë heard his voice and was silent.  And the voice of Ilúvatar said to him. 'Why hast thou done this?  Why dost thou attempt a thing which thou knowest is beyond thy power and thy authority?'" ~The Silmarillion, chap.2 (Of Aulë and Yavanna)

 

And when Aulë realized he had gone to far, he repented and prepared to destroy that which he had made.

 

"Then Aulë took up a great hammer to smite the Dwarves; and he wept.  But Ilúvatar had compassion upon Aulë and his desire, because of his humility; and the Dwarves shrank from the hammer and were afraid, and they bowed down their heads and begged for mercy.  And the voice of Ilúvatar said to Aulë: 'Thy offer I accepted even as it was made.  Dost thou not see that these things have now a life of their own, and speak with their own voices?  Else they would not have flinched from thy blow or any command of thy will.'  Then Aulë cast down his hammer and was glad..." ~The Silmarillion, chap.2 (Of Aulë and Yavanna)

 

And so Ilúvatar allowed the Dwarves to endure, but under a condition.

 

"'But I will not suffer this: that these should come before the Firstborn of my design, nor that thy impatience should be rewarded.  They shall sleep now in the darkness, under stone, and shall not come forth until the Firstborn have awakened upon Earth...'" ~The Silmarillion, chap.2 (Of Aulë and Yavanna)