That was introduced in a patch following Stormblood's release, but was incorporated into the base game, so is available in ARR (though starting late in ARR, so people are more likely to reach Heavensward by the time they unlock them).
The requirements for unlocking command missions begin with becoming a second lieutenant in your grand company. That requires a level 47 dungeon, Aurum Vale, which is the last level specific requirement, so is the lowest level at which command missions can be unlocked.
Once you become a 2nd Lt. and do the pseudo-quest* to introduce squadrons, each time you complete a challenge log entry, there's a chance that an NPC will apply to join your squad. You need to gather a squadron that way, and then level them up by sending them out on training missions. Once they complete a flagged mission which advances the squad to rank 2, you can do command missions with them, where they accompany you into a dungeon.
*(I'm calling it a "pseudo-quest" because the NPCs at your GC headquarters that you need to talk to are marked with a quest icon, but it doesn't have a quest name or show up in your journal.)