They are proof that you've done a certain number of propostions of the treasure and discovery arc. Level 8 is the highest. Propositions are pretty cool, since you'll collect a lot of old scenes and treasures from other FF games. Unfortunately only normal character classes can do propostions. I.e., Rad, Alicia, and Lavian can do them but none of the other special characters can. This makes it a problem to do later in the game. If you really want to do this, Make sure to try and get done with most of them by early Chapter 4. As for the propositions themselves, always send three characters, as I've usually got failed mission with 1 or 2. Also make sure that you send the appropriate class. If the proposition says they need a Chemist, send a Chemist. Oh yeah, Brave and Faith affect things too. In salvage and mining missions, always send a high brave monk, they almost always come back with good treasure or a lot of gil. In discovery missions, a high faith will allow you to find new places. Every city has up to eight or nine propositions. The easiest way to get them over with is to take them on, then go to a spot with two blue dots and move back and forth until the time is almost up then go back to the inn where you sent them out. This way you'll avoid many harsh random encounters bereft of three of your companions.