: Michael Martinez recently posted a good essay on this which he might repost.

: But briefly - the power of the Nine, Seven, and Three Rings was the same (in qualitly if not quantity): to slow the fading of the Eldar, thus allowing them to remain in Middle-earth.

: However, Sauron, who assisted the Gwaith-i-Mirdain in making the Seven and the Nine designed the Rings to be traps. So that he could make a Master Ring and enslave the wearers of the other Rings.

: Celebrimbor alone made the Three Rings (after Sauron had left Eregion), however he used the same "specifications" as for the Seven and the Nine. Thus the Three were also traps, even tho Sauron never actually handled them.

: After the Seven, Nine and Three were made and in use (on the fingers of various notables in Ost-in-Edhil, although who had which Ring is nowhere recorded), Sauron withdrew to Mordor and, as planned, made the Master Ring to control the others.

: *** One Ring to rule them all, One Ring to find them,
: *** One Ring to bring them all and in the darkness bind them.

: However, Sauron failed. When he placed the One on his finger, the Elves became aware of his plan and tore the Nine, Seven and Three from their fingers, thus escaping Sauron's trap.

: Why did Sauron fail? This is an intriguing question which Tolkien nowehere addresses (as far as I am aware).

: It has been suggested that Sauron did not intend Celebrimbor to make the Three Rings. And that if Sauron had only had to ensnare the wearers of the Seven and the Nine he would have succeeded. However the unexpected addition of the Three to the net meant that it was unable to hold all the fish, and they broke loose.

: This is only speculation, but it is a possibility. It may be that Sauron simply underestimated the strenght of will of the Gwaith-i-Mirdain.

: The Three Rings had healing properties that do not appear to have been shared by the Seven and the Nine. These were no doubt added by Celebrimbor, but the basic purpose of the Three was the same as for the Seven and the Nine.

: The above theory (about the Three) may also explain why some thought the Three would be set free with the destruction of the One. Whereas this was never suggested of the Seven or the Nine.

I can see one disparity between this theory where the Seven and the Nine were intended for use by Elves, at least initially, and the incantation associated with the One Ring. The incantation runs:

"Seven FOR the Dwarf-lords in their Halls of Stone"

"Nine FOR Mortal Men Doomed to Die"

The use of the word "For" implies that the rings were intended for the use of dwarves and men from the outset. Alternative wording like the substitution of "To" or "Taken by" or "Used by" instead of "For" would be more in keeping with the explanation of the rings use given above. It should be remembered also that lesser rings not counted in the Three, Seven or Nine were also produced by the Eregionite smiths.