Shanten measures a hand's distance to tenpai: the minimum number of tiles you must swap to reach a hand ready to win. A 1-shanten hand is one swap from tenpai, a 2-shanten hand two swaps, and so on.
Shanten is the compass of efficiency. Each turn you try to lower your shanten while keeping the best ukeire (the count of tiles that advance the hand). Comparing two discards often comes down to comparing their shanten, then their ukeire.
Computing shanten by hand is slow, because a 13-tile hand decomposes many ways; rules engines compute it by standard decomposition. On Tenpai.gg the shanten and ukeire of every move are shown to train that judgment.