I've noticed that many of the forum-dwellers here on GOG have some kind of special title under their avatars, such as Catshade's "The Cat," codagh's "Freak Out!" and michaellung's "I have your file." I'm a bit curious about how these titles are acquired; I don't see any obvious way to enter one on my profile page, so I ask you, how do you get a title? Does it have to be awarded? Do you request it? Is there an obvious big red "give yourself a title" button that I've somehow missed until now?

Just click 'My Settings' on the top of your forum page.

From the FAQ:
How do I get a title next to my Avatar?
Just go to any forum group and click on the My Settings link. Type what you would like in the Forum "title" box, then click "Save and Go Back".

To expand on that, once you've clicked the "My Settings" link (looks like a light switch with the text "My Settings" next to it) you'll wind up here: http://www.gog.com/en/forum/mysettings
Underneath the heading "Personal stuff" on the left-hand side, you'll see your avatar, reputation and a text box labled 'Forum "title"'. Simply enter your title in that (up to 16 characters long).
Then click on the big green "Save and go back" button on the bottom right of the page.

And if i may expand even a little further on that, make sure once you do locate the approriate text field that 4 people already told you where to find even if they each used slight variations of the same exact answer, that you do pick up a 'tastefull', 'non-offensive in any way' board tag'.
For instance the board tag 'WANTS MOAR TITS' while perhaps amusing to some might indeed hurt the susceptibility of some of your fellow board users, namely female board users and male board users who have TITS. And before you ask, yes, they're out there.
So my advice would be to go with an all around family friend board tag like, 'WUV WABBITS', 'APPASSIONATA RLZ' or something along those lines.
Naturally if there wasn't a 16 char limit on the tag, something that the 4 people who answered your question all forgot to mention but still a very important piece of information nonetheless, my advice for a nice board tag would be :
"I think Halo is a pretty cool guy. eh kills aleins and doesn't afraid of anything..."

Uh....16 letters maximum. Also, consider using something as an inside or pop culture joke.
Mine is quite obvious.

Yeah, we're a classy community here, so "Requests more mammaries" is more appropriate.

Alas, the 16 char limit won't allow it :)

I think you'll find, if you re-read my answer, that I didn't forget that at all. Also note that my post has not been edited, so there can be no claims that I later added that important information into my post =)

"I think GOG is a pretty cool guy. eh sells old games and doesn't afraid of anything..."
My awesome title next to my avatar is only funny if you know who the avatar guy is.

HAHAHA, you're right bansama. That one's a massive fail on my end. That's what i get for mixing up booze and the interwebz. That and a massive headache.
Oh, and i would never question your word mate, post edited or not ;)

I know, that comment was more sarcasm on my part.

"Req's > mam'ries" does fit however. It might not be exactly the same, but the gist of it is there.

I was considering editing my post cause now that i can actually read it, it's a bloody mess. But rewriting history is for wusses. I'll leave it as a warning of the dangers of drinking and posting and live with the shame instead.
Thanks for being such a good sport bansama.
Somwhow it doesn't pack quite the same punch :)