In Utah water rights are not necessarily attached/appurtenant to the land. So if you are buying or selling land you want to know what water rights you own or will own and if it is sufficient for your intended purpose.
Water rights are measured in acre feet which is literally an acre a foot deep in water. This equates to about 325,000 gallons of annual use.
For a domestic type use 1 acre foot of underground water is sufficient for a full time residence with a small yard/garden and a couple of large animals like cows or horses.
Farming is much more water intensive and requires sometimes 4-5 acre feet of water per acre that you want to farm.
If you need additional water, in general additional water rights can be purchased on the open market and are listed for sale much like land.
Water rights are treated like real property and are transferred via deed which is recorded with the county recorders office. In addition a report of conveyance is filed with the state engineer
Water shares are shares of stock in a water company. Each share would represent a certain amount of water you are able to use.
More information can be found at http://www.waterrights.utah.gov