A Crucible Commendation is a valuable unit of currency, acquired from a Crucible/Faction rank-up reward package. These packages are sent to the Postmaster each time your Crucible/Faction reputation reaches a new rank (starting at Rank 3, then every 2500 reputation earned).
Crucible Commendations are used in combination with Crucible Marks to purchase Legendary Crucible/Faction weapons, helmets, and chest armor. Commendations are not required for Crucible/Faction gauntlets or boots.
Note: It worth restating that Faction rank-up packages do include a Crucible Commendation.