I get that:

1st party - company making games for their console
3rd party - company making games for other consoles

But a 2nd party I don't get

I read on here that it's like a company owning a portion of a company, how does that work?

the problem is that 2nd party is not an official terminology but something made up by fans but basically people use 2nd party to refer to a number of different things.......

a 2dn party title is a title where a console owner hires an outside studio to develop a game (for example Luigi's Mansion Dark Moon, Nintendo paid NextGen an outside company to develop game, is different than just a publishing deal because Nintendo knew the game they wanted and simply paid an outside company to develop it instead of developing it themselves)

some people call 2nd party studios to studios that are only partly owned by the console maker as was the case of Rare and Nintendo

Does the company only ever get approached by a console owner or, is there some sort of application-esque thing to get to develop a title that a console owner wants

It differs by game. The only sources we know of are developer interviews

Okay, thanks!