Shanten
Shanten 「向聴」 is the number of tiles needed for reaching tenpai. It can also be viewed as the minimum number of tile draws or tile calls required to attain tenpai.
Shanten counts
The lower the shanten count, the closer the hand is to tenpai. Naturally, that is advantageous as far as tile efficiency is concerned. After all, the sooner the hand gets to tenpai, then the sooner can the hand win. For every shanten count, the hand needs any needed tile just to reduce that count.
Iishanten
Example: 1-shanten
Needs: Any 
to tenpai.
Ryanshanten
Example: 2-shanten
-
Needs: Any to 1-shanten.
Sanshanten
Example: 3-shanten
-
Needs: Any 
to 2-shanten.
Open vs closed
Often, the decision to open the hand rests on the tiles needed and the yaku most conveniently acquired. In addition, the shanten count may play a factor too, especially at 1-shanten where a discarded tile could put the hand into tenpai.