There's a limited number of reasons why you wouldn't be able to enter a duty finder:

1. Any members of the party are a class ineligible for the content (Chocobo companions, or anyone currently in a crafting/gathering class)
2. Any members of the party are currently offline
3. A member of the party has not unlocked the duty (You must ensure the duty has been added to each member's duty finder list)
4. A member of the party is too low of level in the class currently selected (Higher levels are absolutely fine; they'll be synced down, but participate just fine. However, being below the level requirement [and, at level 50, the item level requirement] is not an option)
5. You are queueing as a partial party with an invalid combination.

#5 is the likely culprit. 4-man duties (Which is everything prior to level 40 and still quite a bit after) have a very strict party composition - one tank, one healer, and two DPS. You can get around this if you are queueing as a full 4-man party, in which case you can bring whatever combination you want. However, if there's only 2/3 people queueing, you have to adhere to this combination for the duty finder to accept you (as it will fill in the rest of the composition). If both you and your friend are Conjurers (and therefore Healers), you'll be unable to queue together because of this, unless you find other players to queue with.