This is something that's been discussed a number of times (recent example), and has now been added by an open-source contribution by @what, who's contributed multiple significant features now, thanks again!
It's totally optional, but if you want to, you can now write a short bio for yourself through the "Edit your user bio" link at the bottom of the Settings page (by the way, I also added a link to the Settings page at the bottom of the home page's sidebar). The bio supports markdown the same as topics/comments, and is currently limited to 2000 characters (about 300 words). I'm not particularly attached to that limit, but it seemed like a reasonable starting point.
If you write a bio,
there will be an expandable "View <username>'s bio" block at the top of your user page okay, okay, I'm convinced that it belongs in the sidebar. You can see an (unimaginative) example on my user page. I'm not particularly attached to this method of displaying it either, so let me know if you think something else might be better.
I don't know if we need particular guidelines for what's appropriate to put in a bio, but please be reasonable about what you use it for. A link to your site or accounts elsewhere is definitely fine, as long as it's not done in a spammy/misleading way, not too much different from posting on the site itself.
Also, I've topped everyone back up to 10 invites again. As always, these are available on the invites page.