Bilbo takes the Arkenstone. He isn't entirely sure why he does it, but he picks it up and knew exactly what it was. He had some idea it might be of use later on.
The Sackville-Baggins's: Lobelia and Otho. They believed that they were entitled to Bag end since Bilbo had lived much longer than they thought was "respectable."
It makes him invisible. In addition to allowing him to avoid detection, he was able to use it to rescue the dwarves on at least two occasions. The ring also gave him long live, allowing him to live...