They were made by Aulë, the smith and craft master of the Valar. This was against Eru's Plan: Aulë had neither the authority nor indeed the power to create other souls (the result of his efforts was a group of what amounted to puppets). However, because he repented his folly at once and because his motives had been good (he desired children to teach, not slaves to command) Eru gave the Dwarves life and made them part of the Plan. The Elves were still to be the "Firstborn", though, so the Dwarves had to sleep until after the Elves awoke.
If, as has been told, only Seven Fathers of the Dwarves were created, how did the race procreate?
In the the The Silmarillion account of the making of the Dwarves, only the Seven Fathers are mentioned. In Letter no. 212 (p 287), however, Tolkien speaks of thirteen dwarves being initially created:
"One, the eldest, alone, and six more with six mates."
Thus, it seems that Durin really did "walk alone" as Gimli's song said.
In any event, the notion of bearded dwarves seems an assumption with fairly firm foundations.
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