Before we take a look a the Mandarin's rings, let's take a look at his dial.

His standard powers include Running Shot, Charge, Phasing/Teleport, #Toughness, Outwit, Combat Reflexes, and Probability Control.

From Marvel.com:
Discovered by the Mandarin in China’s “Valley of Spirits,” the Ten Rings of Power are actually the product of the dragon-like race Axonn-Karr (or Makluans) from the planet Maklu-IV. The Axonn-Karr utilized them mainly as the power source for their interstellar craft’s engines. The rings’ functions currently cannot be explained by modern Earth science.

The Mandarin learned how to utilize the rings for own personal use and make them respond to his personal commands. Over the years, the Mandarin’s mind has actually come to inhabit the rings so that now no one else can make use of them. The Mandarin can also now control the rings over large distances.