If you are a full year Utah resident, you have income taxed by another state and Utah when you work or otherwise earn money in another state and pay income taxes to that state.
Part year resident's income is usually only taxed by one state at a time. For example, if you lived in Utah for three months then moved to Colorado for the rest of the year and didn't earn any Colorado income before you moved from Utah, you wouldn't claim a credit for taxes paid to Colorado on your Utah return because none of your income was taxed by both Utah and Colorado.
However, if you worked for three weeks in Colorado earlier in the year before you moved there, the wages you earned for those three weeks are taxed by both Colorado and Utah. In this case, you can claim a credit for taxes paid to Colorado on those wages on your Utah return.