I seriously couldn't find a lot without seeing screenshots SO we will have to make do with what I could find. Also you can click on any of these pictures to make them larger, if the resolution is good enough for that.
So, we're starting with the female Avatar. And yes she is customizable so she may look slightly different in each picture, but she should always be wearing tactician/grandmaster gear unless she's reclassed.
Her default name is Robin, but I've changed it each time (why not?). I have currently made Xion from Kingdom Hearts, and I'm developing her story in the Fire Emblem world.
The Avatar, whether male or female, is the mortal vessel of the fell dragon Grima (god of destruction and so on). They just don't know it at first. Some weird timeline stuff happens and suddenly there are two "Grima" vessels - you, the avatar, and the Grima vessel that came to your timeline. The latter becomes Grima and tries to force you to reunite with him/her, so that Grima's power can be complete. Then your friends intervene just when you're about to lose hope and you defeat Grima and blah blah blah.
Left: "Return to Me"
Right: I don't know, actually. But that's Grima up in the corner.
When you finally defeat Grima you have the choice of letting Chrom strike the final blow (which will only seal Grima away for more centuries) or striking it yourself - which means Grima would be utterly destroyed by his/her own self. But that also would kill Robin, unless his/her ties and bonds to other people's hearts were strong enough (and spoiler alert: they are).
And here is Chrom and the female Robin, which I wholly support in shipping. :D Chrom and Robin have a real close relationship, whether male or female Robin is chosen (for now every time I say Robin assume it's the girl unless I say otherwise). If Chrom marries Robin, well, the relationship just gets much closer and also the storyline is much more touching: it's a family that struggles to keep their bonds close through defying destiny.
Some silly shots
The whole gang!
By the way, I really recommend Awakening if you're a Fire Emblem fan. Even if you're not you should give it a try.