Your Twitter Avatar is your profile photo that shows up next to every tweet you post. (You also get a header photo that shows up on your profile page.)
If you happen to be looking a little up and to the right in your profile photo, it will always look like you've captivated yourself with your wise words.
When people click on your avatar, they'll be able to see a bigger version of the image, if one exists, so be sure to use a nice big photo, up to 2MB.
When installing it, you get to choose how to scale and crop it to the 48 by 48 pixel standard size. Keep in mind that app developers have the ability to pull in many different Twitter profile picture dimensions of their choosing.