So mainly to control the nine Nazgul, Galadriel, Agent Smith, and Saruman?
Mainly, but there are a lot of minor abilities.
It could, for example, change a person's appearance. For example, not even wearing the Ring, merely carrying it, Samwise appeared to an orc as a powerful warrior wreathed in shadow, carrying some item of power and menace. Similarly Frodo draws upon this power on Mount Doom to intimidate Gollum, appearing as a figure robed in white, bearing a wheel of fire (presumably, the ring has a mighty ego, and so always includes itself in the imagery.
This may be an extension of a more general ability - an enhancement to the bearer's ability to dominate others, ring or no, and the images described may be more mental illusion. Sam and Frodo have little such ability natively, so the enhancement doesn't mean much.
It is repeatedly said that a goodly chunk of Sauron's power is in that ring. As angelic beings go, Sauron was of higher order then Gandalf and Saruman, so we could be talking about a *lot* of power in there. Surely, that can't all be required to dominate other wearers of rings. The way Gandalf and Galadriel speak, the ring also would represent a significant boost to their personal power - presumably, they know enough about magic to access the power Sauron had locked away in there, in ways that mere mortal hobbits couldn't.