The Three Elven Rings are noted in Of the Rings of Power and the Third age as having been made last, and that Sauron had no part in their making (despite which they were still subject to the domination of the One):
Now these were the Three that had last been made, and they possessed the greatest powers. Narya, Nenya, and Vilya, they were named, the Rings of Fire, and of Water, and of Air, set with ruby and adamant and sapphire; and of all the Elven-rings Sauron most desired to possess them, for those who had them in their keeping could ward off the decays of time and postpone the weariness of the world. But Sauron could not discover them, for they were given into the hands of the Wise, who concealed them and never again used them openly while Sauron kept the Ruling Ring. Therefore the Three remained unsullied, for they were forged by Celebrimbor alone, and the hand of Sauron had never touched them; yet they also were subject to the One.
So that's the Three, and that's how they were significant, but what of the others?
An important thing about the others is that's there's actually nothing in Tolkien to indicate that the powers of the nine are any different to the powers of the Seven; rather, what differentiates them is the species they were given to, with a Ring of Power having a different effect on a Man to the effect that it has on a Dwarf.
Regarding the number given to each species, there is a note in the Galadriel and Celeborn material in Unfinished Tales as follows:
There Sauron took the Nine Rings and other lesser works of the Mírdain; but the Seven and the Three he could not find. Then Celebrimbor was put to torment, and Sauron learned from him where the Seven were bestowed.
So the Nine Rings that Sauron gave to Men were just the first nine that he had captured, and after having found where the remaining Rings (excluding the Three) were located, those were the ones that he gave to Dwarves.
Therefore the numbers that he gave to each of Men and Dwarves appear to have been based on the order Sauron captured them in - first the Nine, then the Seven - and not much more.
There's further room for some interesting speculation which it's probably worth mentioning to ward off follow-up questions. We know that there were three houses of Elves and Seven houses of Dwarves, which match with the numbers of Rings each had. Were there also nine houses of Men (in total, including the three of the Edain that we know of for certain)? There's nothing in Tolkien to suggest this.