What, exactly, is the difference?

I know that a first party is owned entirely by one of the console manufacturers, and second party is partly owned or contractually tied to the developer. But what does this mean in terms of freedom? Can a second party developer split from a manufacturer, like Bungie? Was Bungie first or second party?

And what is it when, like Mass Effect, Microsoft takes a third party company and publishes the game? two and a halfth party? Or was Bioware second party as well?

Please help me clear this up.

Naughty Dog is first party, right? And Insomniac and Sucker Punch are second party.

According to Wikipedia, BioWare is a subsidiary of EA, so it's third party.

So how did Microsoft get that deal, and how did they lose it?

Anybody know?

Bioware was a independent studio. MS published Mass Effect so it's considered a 2nd party title as it was not a MS owned studio (but people still consider it first party as it is an exclusive to the console) . EA bought Bioware so Mass Effect 2 will be a 3rd party title, unless there is some co publishing deal we don't know about