Arya is the middle child of Eddard and Catelyn Stark; she's younger than Robb and Sansa (and also Eddard's illegitimate child, Jon Snow), but older than Bran and Rickon. Arya gets taken to King's Landing when her father goes to be Hand of the King. And although she fights constantly with her sister Sansa, Eddard gets Arya lessons in sword-fighting with Syrio Forel. As it turns out, her lessons with Syrio save her life when the Lannisters take over. Arya escapes and lives on the streets until she is found by Yoren of the Night's Watch. What happens after? Guess we'll have to read the next book.
Arya is a total tomboy, the little girl who would rather play with a sword than marry a knight. (This is confirmed through her name choice for her wolf, Nymeria, named after a historical warrior queen.) Sure, she makes trouble sometimes, like when she finds a secret tunnel that leads outside the castle and she needs someone to let her back in (33 Arya 3). But she's also brave enough to go after what she wants (sword-fighting lessons) rather than what other people think she should have because she's a girl (sewing lessons).
Of course, let's make two things clear: (1) there's a thin line between being an independent individual and just being a jerk. Eddard helps Arya see this in relation to Septa Mordane: Arya says she hates Septa Mordane, but Eddard points out that the septa is just doing her job and that Arya needs to find some way to deal with their disagreements other than just running off (23 Arya 2.51). To her credit, Arya starts being more thoughtful toward both Septa Mordane and Sansa. Basically, she continues to be a wild child, but apologizes more.
And (2) Arya's independent streak only works out for her because she has a loving family who helps her become the fighter that she wants to be. First Jon gives her a sword and then Lord Eddard gets her sword-fighting lessons (probably through Groupon). It's pretty easy to imagine things not working out for her if her family weren't so awesome. For instance, compare her story (loving support from her dad) to Samwell Tarly's story (threats of murder from his dad). So yeah – you're one lucky tomboy, Arya.
Besides her independent streak, there's a lot to like about Arya: she's smart (11 Jon 2.79); she's adventurous (all over the book) and brave (check out the story of her standing up to and punching a "ghost" who turns out to be her half-brother [51 Arya 4.101]); and she's friendly with people of lower social status (just like her father Eddard, Arya likes to talk to common folk [16 Sansa 1.28]). Of course, that last part doesn't work out so well for her one commoner friend, Mykah, the butcher's boy. (See below for more on that.)
We like Arya as a character, but we also like reading her chapters because we're never sure where they're going to go. Of all the Stark children, Arya has perhaps the strangest story arc and gets into the strangest situations. For instance, in one chapter, she uncovers two serious conspirators (Varys and Illyrio Mopatis?) talking about possibly killing her father (33 Arya 3). Whereas Robb ends up going to war and Sansa ends up as a hostage – both pretty typical happenings for noble kids – Arya ends up hiding out as a commoner. To be completely honest, Arya is always surprising us. She's the wild card of the bunch.
And, reality check, she's all that at nine years old (8 Arya 1.43).
Mykah is a servant, the butcher's assistant/apprentice. (We think that's what "butcher's boy" means, but maybe he's just the butcher's son?) Mykah is just a male friend of Arya… until one thing leads to another (see 16 Sansa 1) and Mykah is killed by Sandor Clegane (17 Eddard 3). Mykah's death doesn't seem to affect anyone but Arya: it makes her realize that the system of justice in Westeros isn't entirely, well, just.