Water rights are rights granted by the State of Utah, through the Utah Division of Water Rights (also known as the State Engineer's Office), that allow a person to use a specified amount of water from a specified source at a specified place for a specified use. For example, the State may grant you a water right that allows you to divert 1 acre-foot from a well to use on your residential parcel to irrigate 0.25 acres. Thus, a water right is granted directly from the State to the water right owner, as depicted below.
With water shares, the State issues water rights to a water company (ditch company, canal company, irrigation company, etc.). So the company is the owner of the water rights. The company issues shares to its shareholders. By virtue of owning shares, the shareholders are entitled to receive and use a portion of the company's water rights.
Many times people say "I have 3 shares of water," when what they really mean is that they own a water right that allows them to divert and use 3 acre-feet of water. Conversely, many times people say "I have 5 water rights," when what they really mean is that they have 5 shares in a particular irrigation company.
The distinction between water rights and water shares is important in many contexts, including how the water rights or water shares are transferred, how change applications are filed, and whether the water transfers automatically with the land when the land is sold.