I think it's fairly well explained what powers were given to the One Ring, Elves, and Humans, but the dwarf rings are little explained in the series. Why were these rings important, or what kinds of powers did they give?
The Dwarves used their Rings to establish their treasure hoards (which in turn attracted dragons). As per Wikipedia: "Gandalf mentions a rumour that the seven hoards of the dwarves began each with a single golden ring".
As far as effects: according to the LotR Wiki:
Sauron, according to portions of the Silmarillion, was unable to force the Dwarven bearers to submit to his will. Indeed, the rings did not even turn them invisible; they were immune to some of the more detrimental of the rings' effects.
It is believed that the dwarves' natural hardiness, and the fact that it was only the more powerful dwarf lords who possessed them, made them resistant to Sauron's control.
The net effect of these rings was to bring vast wealth to the wearer and cause him to become extremely greedy.
At the time of The Lord of the Rings four rings had been consumed by dragon fire and the rest acquired by Sauron. The mightiest of the Seven was taken from Thráin II, heir of Durin, who had been captured, imprisoned, and tormented by the Necromancer (Sauron in disguise) in TA 2845.
A Man who gained possession of one of the Seven likely would have become a wraith, just as those who wore the Nine did.
The only power over [Dwarves] that the Rings wielded was to inflame their hearts with a greed of gold and precious things, so that if they lacked them all other good things seemed profitless, and they were filled with wrath and desire for vengeance on all who deprived them. But they were made from their beginning of a kind to resist most steadfastly any domination. Though they could be slain or broken, they could not be reduced to shadows enslaved to another will.
(Note that it is the Dwarves' “biology” that gives them this resistance, this isn't a property of the rings themselves.)
The Silmarillion includes a chapter "On the Rings of Power and the Third Age". I think that's the most information we get on what the Rings actually do.
The Dwarves indeed proved tough and hard to tame; they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows. They used their rings only for the getting of wealth; but wrath and an over-mastering greed of gold were kindled in their hearts, of which evil enough after came to the profit of Sauron. It is said that the foundation of each of the Seven Hoards of the Dwarf-kings of old was a golden ring.
I think its important to note that the rings were not Dwarf rings, any more than the nine were Human rings - all of these were forged by elves but touched in some part by Sauron. They could have all had exactly the same powers.
What we do know from the text is what those 16 rings brought out in those who used them, comparably the effects on humans and dwarves, and that none of these lesser rings were offered to elves. We have no way to know what effect the three rings would have had if they were worn by humans, dwarves or hobbits.
Look at the effects of the 16 - they enable things highly desirable to the receiver. Humans desire unending life (the Downfall of Númenor), and dwarves dream of great wealth. Their desires are fulfilled, but in an awful way since they were tainted by Sauron. For the immortal elves, that desire is for their realms to be as untouched by time as they are (this worked very well in Rivendell and Lórien with the rings of Elrond and Galadriel).
Now I am speculating here, but I suggest that the lesser rings lacked the sophisticated elemental personalities of the three rings, much as the One Ring appeared as a lesser ring because it lacked any real markings; I believe all of these lesser rings shared the same powers.