UZR quantifies a player's entire defensive performance by attempting to measure how many runs a defender saved. It takes into account errors, range, outfield arm and double-play ability. It differs slightly from DRS (Defensive Runs Saved) in its formula, but the concept is the same.
UZR uses Baseball Info Solutions data to chart where each ball is hit. Say, for instance, a center fielder sprints to make a nice catch on a fly ball. Then, say data from BIS tells us that similar fly balls get caught 60 percent of the time. That center fielder gains, essentially, 0.4 bonus points for difficulty. If he can't make the play, he loses 0.6 points. At the end of the day, that player's overall score gets adjusted to the league average -- and then that score gets adjusted for how many runs the once-adjusted score is worth.
Why it's useful
Errors and assists barely scratch the surface of what makes a successful defender. UZR exists to help better value defenders on elements such as range, efficiency on double-play chances and first step.
Looking at a team's collective UZR can help forecast whether a pitcher will be helped out by his defense. More specifically: For a fly-ball pitcher, it might be worth looking at his outfield's UZR. And for a ground-ball pitcher, one could take a look at his infield's UZR.