This building is known as Council Hall, it is now a State Visitor's Center and Travel Counsel. It was originally Salt Lake's City Hall which was then located at 120 East and 100 South in Salt Lake City. It was relocated in a joint effort and reconstructed on 300 North, across from the Utah State Capitol.
As you can see it is constructed of Red Sandstone. This is another of the buildings that I am glad to see was saved from the wrecking ball!
The information plaque for Council Hall