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.
It doesn't change the story in any way but it does change the amount of enemies and the location of them on missions while making them significantly stronger and with different items + skills. The only way to get different storyline is to change your gender during another playthrough and have the characters marry different people. Please note that I haven't finished my Lunatic playthough but haven't seen any differences so far. Also, this is expected as the script writers had TONS of stuff to write already (most of which you will never be able to read), so this is most likely just a gameplay difficulty.
Spoiler:
Lucina will have different dialogue with you if you are married to her throughout the game. You can also marry Chrom which will produce a slightly different story.
Best Answer
Anna doesn't die so much as retreat. Even if she retreats and the Barbarians take the item from the village, it has no bearing on the next time you meet. That's where it pays to keep her alive (and relatively out of harm's way, if possible).
While she's alive in your current mission, she lowers the difficulty by acting as a fantastic decoy. Keeping her in that state is a much more difficult proposition, but based on the conversation with her next time you meet, your actions in this mission have no effect on her joining or not.
Based on her character page on the Fire Emblem Wiki (warning, lots of spoilers), she won't join until the next time you meet, even if you manage to keep her alive this time.