Below are some of, in my opinion, the best articulated descriptions of Rainbow Dash from various fan stories I've read. I sometimes borrow from them because they are just so awesomely on target.
“Dear Princess Celestia,
when you have strong friends, it’s hard to see them at a weak moment. But it’s exactly the ones who hold us up, the ones who show us how to be strong, who need our help the most. When they fall, they fall harder under the weight they’ve been carrying for us.
I learned today that even the strongest hearts can be stricken by doubt, even those that carry their friends to greater heights. And I learned that when this happens, it is our duty as a friend to carry them as they have carried us. I know my friends will always be there to catch me if I should fall, and I’m honoured that I can return the favour.
Your Faithful Student
“A Dash of Gold,” Chapter Two
A lot of ponies would be surprised to hear what a good boss Rainbow Dash is. She’s lazy, impatient, hot-tempered, stubborn, and thinks everything is a competition — not exactly the traits one looks for in a leader. However, she makes up for all that by having buckets of charisma.
Now, she’s not some kind of charming sweet-talker like most ponies think of when they say that a pony has charisma. Honestly, when it comes to most of her social skills she’s not that great. What the boss has going for her is this kind of personal magnetism that makes you believe in her, believe she can do all the crazy things she says she’s gonna do. Then she goes and sets some kind of impossible goal and pretty much everypony thinks she’s crazy, but you stick by her because deep down you know that she just might be able to pull it off. Then she goes and does it, and you know why you’ve been following her all this time.
Because to a pony like Rainbow Dash, ‘impossible’ isn’t so impossible after all.
Pretty much everypony on the weather team was in awe of her. It was hard not to be. More than that, we were all fiercely loyal to her. Maybe that’s why she got the Element of Loyalty; she’s not just personally loyal, she inspires that kind of loyalty in others.
“The Life and Times of a Winning Pony,” Chapter 6
It was completely insane. Common sense said that there was no way Rainbow’s plan should work—the numbers just didn’t add up. I saw Twilight shooting a look my way, and I could take a guess at what she was thinking. I could shut Rainbow down if I really wanted to—it’d be messy, but I’d be within my rights to challenge her on this. She was making a completely irrational decision that put ponies at risk of injury, or worse, chasing a plan that simply couldn’t work. The smart thing to do, the responsible thing to do would be to put a stop to this.
Except ... I believed in Rainbow Dash.
Faith can be a crazy thing like that. Rainbow wasn’t some goddess or even some kind of honored ancestor, she was just a normal pony with (if I’m being honest) some pretty big character flaws. She was suggesting a crazy plan that math and common sense said she just couldn’t do, but since when has Rainbow let something like that stop her?
Our friendship had gone through all kinds of hell, not just over the past few days, but for a pretty big chunk of our lives. I’d spent a couple years hating her guts after Flight Camp, and there had been plenty of times when things had gotten rocky on us. Despite all that ... I believed in her.
I believed that she just might be able to pull this off after all.
We all took back to the air with a little extra something pushing us. So we couldn’t pull it off when everyone gave a hundred percent? Well then—I know it sounds horribly cheesy to say it, but we would just have to give more than a hundred percent. Rainbow ... kinda has that effect on ponies.
“The Life and Times of a Winning Pony,” Chapter 13
I’m fluent enough in Rainbow Dash-ese to know that meant ‘yes.’ I gave her a bit of time to build up her courage and get comfortable—getting Dash to talk about anything that was bugging her was an exercise in patience. She wasn’t timid like Eepy or prone to letting things bottle up like Blossom could be, but she could be stubbornly independent-minded. It could be a lot of work to get Rainbow to admit that there’s something bothering her that she can’t handle on her own.
The key was to let her set her own pace working up to a confession. Badgering her would just make her set her hooves and start getting defensive. Instead, I let her approach the issue on her own terms. “So,” Dash began, “I was thinking...”
...You have to feed Dash’s ego a little when she’s being all nervous and lacking in confidence, or else she’ll just completely crash on you. Not too much, though, or she’ll start getting a big head. It’s a delicate balancing act, but I’ve got plenty of experience dealing with her.
...After a minute or so Rainbow let out a sigh, and pulled back a bit. Not completely out of the hug, but she put a little bit of distance between us to let me know she was okay now. “I just—I have this really bad habit of screwing things up with my friends. And this...” Rainbow’s eyes dropped down to intently study my sofa. “If I screw things up with Pinkie Pie, I don’t know what I’ll do.”
I put a hoof under her chin, and brought her her eyes back up to meet mine. “Rainbow, the past is the past. You won't mess this up. Besides, even when you do screw up, you always fix things. You fixed things with me, after all. Compared to that, you and Pinkie will be fine.”
“The Life and Times of a Winning Pony,” Chapter 14
“Ah, young one, such empty courage,” the ape mused. “There is much strength in you, more than I have ever seen in any creature before. Why hide it behind false bravado?”
“The Days Just Don’t Get Any Shorter,” Chapter 1
Please, Ms. Dash was my mother," I insisted, trying my hardest to soften my voice. A lot of ponies say my voice is... unique for a mare. What they mean by that is it's grating. I was pretty self-conscious about it when I was a kid and even sometimes... Ugh! This isn't about me! Getting back to Mac... "Rainbow is fine."
“The Sky is Falling,” Chapter 4
Twilight's eyes grew even softer, if that were possible. "Dash. Just because you're the Element of Loyalty doesn't mean you can't expect it from the rest of your friends."
“When Fears Are Shed”
“Relax Derpy, Dashie’s not mad at you either.” Pinkie reassured the grey pegasus. “Dashie’s just … well, she’s not good at saying things sometimes, and she ends up not saying what she really means.” Pinkie let her gaze drift over to Rainbow for a long moment, before turning it back to Derpy. “She’s not mad at you, she’s worried you might get hurt. Dashie can’t just come out say it though because she’s not good at saying stuff like that, so she gets all twisty and mixed up. Dashie is a silly pony.”
“My Little Denarians,” Chapter 12
I guess that meant Rainbow was crazy, but it had been a kind of crazy we’d needed. Well, that’s not entirely right: Rainbow was plenty scared, it’s just she buried all that fear under a thick layer of bravado and seemingly endless self-confidence.
That had been enough for the ponies, though. Fear is one of those emotions that’s easily influenced by the people around you. If you look around and see that everyone else is scared, you’ll start getting scared too. It’s pure instinct; your animal brain is going to start screaming that anything bad enough to scare everyone else is something that you should be scared of too. That’s why a huge chunk of combat training is about controlling fear – if one person starts panicking, it makes everyone else around them more likely to start panicking too. Just look at how any large group of people can transform into a terrified mob in the space of a couple seconds.
But bravery can work exactly the same way. If one person stands up against the bad guys and says they’re not so tough, then it gets easier for everyone else to find their courage. That’s what Rainbow Dash had done for us. More than any of us, she’d believed that we were going to win this, that we would all get out of this in one piece and get a happy ending. She believed it so hard that the rest of us had started to believe it too.
“My Little Denarians,” Chapter 16