Is it the publisher or the devloper?

I see Mario and Sonic being cosdered 3rd party.

Yet Mario Party is considered 1st party.

Though Mario Party is developed by Hudson Soft and Mario and Sonic is Published by Nitnendo in 2 of the three main territories.

So which is it?  (Or is Hudson Soft 1st party and i don't know it?) 

(Using Nintendo as an example):

-1st-party if Nintendo themselves makes it (Mario Galaxy)
-2nd-party if Retro Studios or HAL Laboratories makes it (Prime or Smash Bros.)
-3rd-party if a company that is not Nintendo or owned by Nintendo makes it (Resident Evil, Red Steel, No More Heroes)

Who publishes it has nothing to do with it. MS published Bioshock, but they do not own 2K Games.

So in otherwords... Mario Party 8 is a giant 3rd party success?

Seems like that games been overlooked then.  Likely due to the whole... Mario in it thing. 

Well Mario 8 is second party and most second party games get filed under first party, because the first party is the one who makes the money from the game and ultimately oversees the project.

What makes it 2nd party then? I thought 2nd party was when the company held a significant part of the stock/company only made games for that system due to an agreement.

I mean Hudson soft is owned by Konami and to my knowledge throws games everywhere.

Who published it has EVERYTHING to do with it. Use common sense: no third party considers Mario Party 8 or Mario and Sonic a big breakthrough win for third parties. They are games made under special circumstances which we might call "second party," but which ultimately amount to first party competition for other third parties.

Here:

http://www.vgchartz.com/forum/thread.php?id=12994

As Erik said, who publishes it makes all the difference.

Using the Nintendo example above, what is the difference between your 2nd and 3rd parties? Who actually published the game.

If N makes and publishes the game, thats 1st party.
If N only publishes the game and some other company is in someway owned by Nintendo makes it, thats a 2nd party game.
**Normally, these two are pretty much considered the same thing. As N is still chief and the games are exclusives.

If N has nothing to do with the creation or publishing of the game and is merely collecting license fees. That is 3rd party.
**This is where N has failed the last two generations and due to the extreme success of the Wii will fare a billion times better now.

 This is a bad example and not the norm. MS basically paid for the exclusivity on consoles for this game. Part of that deal was full financing and publishing. Essentially Bioshock could be considered a 2nd party game while 2k is definitely a 3rd party company, as they are not financially bound to any one company like Rare or Retro Studios.

First party is where the console company develops it in their own studio.

Second party is where the company gives the project to a wholly owned subsidiary, i.e. a separate corporate entity that could in theory split and join another company, example: Bungie under MS, Rare, Monolith Soft. For games like Mario Party, where the IP is all first-party and it is also published by the first-party, it is effectively second-party as the real developer is "invisible' as in a true second-party. It's like subcontracting a job: you still own it.

Third party is developed by an external company.

If N makes and publishes the game, thats 1st party.
If N only publishes the game and some other company is in someway owned by Nintendo makes it, thats a 2nd party game.

Second party is where the company gives the project to a wholly owned subsidiary, i.e. a separate corporate entity that could in theory split and join another company, example: Bungie under MS, Rare, Monolith Soft.

Careful. If a company is WHOLLY owned by a first party, they are part of the first party. If the first party owns at least 51% of a company, they control that company, and it is first party.

Bungie was wholly owned by Microsoft. They had no power to break away from MS on their own; MS let them buy themselves back.

Rare was independant from Nintendo. N owned 49% of Rare, and Rare's founders owned the other 51%. Microsoft bought the 51% first, and then Nintendo sold the other 49%. So Rare was "second party" to Nintendo, and are now first party to Microsoft.