mIRC can be downloaded here: http://www.mirc.com/
Go to Tools > Options.
Select "Connect" on the tree on the left.
Enter your nickname exactly as it appears on twitch.tv. If your nick on mIRC is mismatched with your nick on twitch.tv, the chat server will not allow you to connect and it will not give you an error message.
http://i.imgur.com/Rhmow3x.png
Select "Connect > Servers" on the tree on the left. Click "Add". Enter irc.twitch.tv in the IRC Server field. Enter your twitch.tv password OAuth password in the password field. Be sure to include "oauth:" in your password otherwise it will not work. Click "OK". Click "Select". Click "Connect".
To set up automatic joining of channels, go to Tools > Options. Select "Connect > Options" on the tree on the left. Click Perform. Check "Enable perform on connect". Click "Add", select "Twitch.tv". Add a "/join" command for each channel you want to automatically connect to. See the above image for examples. One command per line. Click "OK". NOTE: Do not add a "/nick" command to these perform paramaters. It may cause you to be unable to connect at all. This is a bug with Twitch.tv as far as I am aware.
If you've done everything correctly, your IRC should look something like this.
UPDATE 9/19/13:
- Use irc.twitch.tv instead of <yourtwitchtvname>.jtvirc.com.
- Change your connect password from your twitch password to your OAuth password (make sure to include "oauth:" in the password box).