I'll give you the detailed breakdown of everything. but the TL:DR is your gonna want an Xbox for NG Black, and NG2.
There are multiple versions of each game that all have their differences.
There is NG 2004, NG Black, NG Sigma, NG Sigma+ --- NG2, NG Sigma 2, NG Sigma 2+ ---NG3, NG3 Razors Edge. As well as the Master Collection versions.
Ninja Gaiden Black is basically the Directors Cut version of NG 2004. It is considered The Definitive version of the game, It takes the DLC additions from NG2004 and organically incorporates it all into the Main game, in NG 2004 Harder difficulties had you fight the same enemies but with larger health bars, NG Black has you fight entirely different enemy types depending on the difficulty.
Ninja Gaiden Sigma is the PS3 version of NG1, it has some additional content such as an additional weapon, Boss, and playable character, and well as Balance changes and Pacing changes. Whether or not these additions are improvements is up to debate. there are pros and cons to everything Sigma changes, but all in all NG Black is seen as the more pure experience, Sigma is more like a Remix.
Sigma Plus is the Vita version The additions here while optional can effect the balance of the game even more, you can fire Arrows in the air now, you get costumes that buff your attack and defense, as well as Accessories that can buff you as well, and this all stacks. Using all of these Sigma, and Sigma Plus additions can take parts that were previously quite challenging in NG Black and completely trivialize it because of how busted the right combo of things can turn out. It can be fun to mess around with, but again Black is recommended to play first. The Master Collection comes with the Sigma Plus version of content.
When it comes to Ninja Gaiden 2 (Xbox), and Ninja Gaiden Sigma 2 these game are entirely different experiences. Again Sigma 2 adds additional Weapons, Playable Characters, Bosses. but this time the changes to the games balance is far more drastic. Xbox NG2 prides itself in throwing overwhelming amounts of enemies at the player, on a Base 360 Slowdown from the shear amount of chaos on screen was common. To avoid slowdown on PS3 Sigma 2 severely cuts down on the enemy count. but due to decreasing the amount of enemies fought at once, this requires tons of balance changes. Ranged Weapons that were made to kill multiple enemies are Nerfed or outright removed, since if they stated the same all nearly all fights in the game would be trivialized by the Windmill Shuriken that would decapartite enemies in Xbox NG2. The are also many other balance changes that are basically just trying to make an easier game, artificially harder such as making enemy grabs practically 1 hit kill the player on harder difficulties. Sigma 2 is not an improvement Xbox NG2 is highly superior, and Sigma 2 is only worth playing if you end up curios about the additional content, which there is a fair amount of at least.
When it comes to the Vita version Sigma 2 plus it added the Ninja Race mode which is five levels that have you race against a decreasing timer that fills back up with kills, and getting a long kill chain increases your damage output, it's meh.
NG3 was straight garbage, Ninja Gaiden 3 Razors Edge was the effort to fix Ninja Gaiden 3, and considering how bad NG3 Vanilla is, Razors Edge did an admirable job trying to salvage the whole game. allot of mechanics were removed in NG3, and basically brought back in Razors Edge, still NG3RE can be a love it or Hate it kind of game still, since it's still quite a bit different from NG1 and NG2, and it has it's own Flow to it's gameplay that can be hard to adapt to for allot of players,
PS3 Sigma 2 and NG3RE had online Co-op modes, in the Master Collection the online part is removed, you still have the same content, but part of it is nerfed so it's more balance for a Single player to beat, this can actually be a point in favor of the Collection because beating the hardest challenges originally required not just One but Two impossibly good players. now it's actually somewhat manageable.