WAR measures a player's value in all facets of the game by deciphering how many more wins he's worth than a replacement-level player at his same position (e.g., a Minor League replacement or a readily available fill-in free agent).
For example, if a shortstop and a first baseman offer the same overall production (on offense, defense and the basepaths), the shortstop will have a better WAR because his position sees a lower level of production from replacement-level players.
For position players: (The number of runs above average a player is worth in his batting, baserunning and fielding + adjustment for position + adjustment for league + the number of runs provided by a replacement-level player) / runs per win
For pitchers: Different WAR computations use either RA9 or FIP. Those numbers are adjusted for league and ballpark. Then, using league averages, it is determined how many wins a pitcher was worth based on those numbers and his innings pitched total.
Note: fWAR refers to Fangraphs' calculation of WAR. bWAR or rWAR refer to Baseball-Reference's calculation. And WARP refers to Baseball Prospectus' statistic "Wins Above Replacement Player." The calculations differ slightly -- for instance, fWAR uses FIP in determining pitcher WAR, while bWAR uses RA9. But all three stats answer the same question: How valuable is a player in comparison to replacement level?
Watch: Billy Burns thrives with the bat, on the bases and on defense, raising his Wins Above Replacement.
Why it's useful
Obviously, the goal of baseball is to win games, and WAR quantifies each player's value in terms of a specific number of wins. A player with a WAR of 0 is essentially a replaceable piece, while a player with a WAR of about 8 should almost always be an MVP candidate. And because WAR factors in a positional adjustment, it is well suited for comparing players who man different defensive positions.
WAR is an all-encompassing evaluation tool, and it can help rank players across positions. But WAR also places an emphasis on defense and therefore isn't the most accurate metric by which to assess a player's fantasy performance.