ORIGINALLY, 2P was an alternative coloring and/or character design for the Nyotalia characters and the main Axis Powers that Himaruya made. These were posted on his website sometime around July 2011. The only one that became noticeably popular at that time was fem!Prussia and her 2P design.
NOW, the fandom has basically butchered that, as usual, and 2P is basically the characters in an alternative universe with more turmoil. Which, is unofficial, by the way. The way this whole "dark-side" thing originated is because Ameriko's character design carries a baseball bat while 2P!Ameriko carries a baseball bat with nails in it, assumed as a weapon. Also, a lot of the 2Ps have red eyes and darker colorings. So people have assumed that that meant they were the characters' dark side.
Speculation is that 2P!England is actually a good cook, as 2P!Igiko holds a tray of edible-looking pastries. My favorite headcanon that I've read, though, is that in 2P world, England is actually a really good cook, but he puts chemicals and poisons in all his pastries and that, assuming America was still raised by him, was why America is so messed up and goes on killing rampages with his bat. Sorry, I read that on tumblr, and I just love that anon's headcanon.
Most of the characters retain the same coloring with minor changes and different outfits and hairstyles, and the like. The most drastic 2P would be that of Ameriko (and America, by default, since there is no official 2P!America), who has dark brown/black hair, red eyes, tan skin, and is practically Ameriko's opposite. I want to point out that many people mistaken 2P!America to have red hair light skin, but the official looks almost Hispanic. Just putting that out there, since it's one of my pet peeves at the moment.
EDIT: Here are the official designs
Himaruya's Official: www.geocities.jp/himaruya/4ka0… | www.geocities.jp/himaruya/4ka0…
Side-by-side Comparison: analmouse.tumblr.com/post/1570…
The links are courtesy of Squircl and analmouse respectively
Hope this was helpful, everyone!