How to verify signed bitcoin messages on my own website? On my website, I want to be able to verify that messages are signed by a certain bitcoin address, similar to how it's done here. From my research it seems I have to use JSON RPC with some sort of verify_message(signature, message, addrtype) function? 

Is this the only method for doing this? And if so, does this require me to install bitcoind or armory on my web server?
You don't need all of bitcoind just for this functionality. Look at the code on the brainwallet.org page you posted: http://brainwallet.org/js/bitcoinsig.js

It seems like you could take this exact code and implement it on the server-side with a JavaScript interpreter such as node.js.

It depends on the programming language you're using but I'm sure this functionality is available in most of the bitcoin libraries: https://en.bitcoin.it/wiki/Software#Libraries