1. Branded phone vs UnBranded phone:  generally this is where they put the "for extra $$" into the menu of the phone and remove or lockout the standard free apps and locations that the mfgr of the phone created.  Take ATT, they put ATT Mall on every phone, buy ringtones in the select ringtone submenu, for pay services on the photo menu, for pay servies on the video menus, etc etc. If you UNBRAND the phone, your phone returns to the base configuration of the mfgr which may be more convenient.  On my SE phone, debranding meant I got A-GPS and Google Maps which ATT had locked out, and the abilty to turn off the 3G only function of the phone which they had locked in. additonally, the bothersome, 'this app wants to connect to the internet' message could be adjusted for each app.

2. Unlocking refers to choice of carrier. Your phone is set to respond to a particular carrier and sim. Your sim is locked to a carrier for the phone to work, they must match. Changing that carrier can be done several ways. You can pay to get the unlock code from a third party or you can get ATT or your new phone company to unlock it for you.  If you are going to switch to T-Mobil, their customer service support people should be able to give you the unlock code for your phone, since they want you to be their customer.  If it's because you are traveling over seas, and you want to use a new sim temporarily, then its to your benefit to use a 3rd party (or ask ATT if they will give you the unlock code) so that you can put another SIM in.  

3. My surmise on the site's device. You put in T-Mobil's sim, and the slim card on the back of that, fools the iPhone's firmware into thinking it's a ATT sim, but since your IMEI for the phone and the sim's serial are on T-Mobil's network and not ATT's it works as if the phone is unlocked/jailbroken.