DRS 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 only slightly from UZR (Ultimate Zone Rating) in its formula, but the concept is the same.
DRS 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
Because errors and assists barely scratch the surface of what makes a successful defender, DRS exists to help better value defenders for their range, positioning and first step.
Looking at a team's collective DRS can often 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 DRS. And for a ground-ball pitcher, you should take a look at his infield's rating.