Support relationships are the bread and butter of Fire Emblem games. When units are within 2 squares of a another unit they have a support rating with, they will gain stats in fights. Sort of like what you get for units beside each other, or using Pair Up. But where the side-by-side boosts increase secondary stats, like To-Hit, Dodge, or Crit Chance, Supports increase primary stats, like Strength, Magic, Defense, and the like. The exact stat boost is dependent on the two units, and is also affected by their current level of relationship. The higher the support rating, the better the stats.
If the support relationship is between a male character (such as Virion) and a female character (like Sully), you'll see that they can hit up to the S class support rating. If they hit that, they automatically get married. There's no veto or "Are you sure?" type of screen. You hit that conversation, they're married. Done deal. So be careful when you're looking at that last support conversation, as it may have consequences you're not planning on.
In previous games, each character could only have up to three supports. In Awakening, the number of supports you can have seems to be unlimited. There's no reason not to fulfill every single support conversation you can, except for the caveat mentioned above.
A sortie party met via StreetPass will stay on the map indefinitely until you Parley with them and either Fight or Shop. However, they will be defeated and leave if a Risen battle appears on their spot and you don't immediately help them out with that Risen battle.
Best Answer
Stat boosts, from either tonics or surges, disappear after one battle, whether the character with those boosts participates in that battle or not.