What is World 3-1?
The name is rooted in video game history.
In the original Super Mario Bros. for the Nintendo Entertainment System, there were eight worlds, divided into four stages each. Three main stages and then a castle. Your princess was not in that castle.
The ending of each main stage usually presented the player with a staircase, which they could climb in order to jump at the top of the flagpole, earning maximum points. As the game progressed, the staircase became more complex, trickier, and the player couldn't assume their victory was secure.
In World 3, Stage 1, a single Koopa was sent towards the player, down the staircase - a wrench in the usual easy ascent to high score stardom. A clever player, knowing that Koopas can be bounced upon multiple times, would see this as a golden opportunity.
By stopping the Koopa at just the right spot, and jumping onto its shell, sending it into the stairstep to the right, you could bounce on it indefinitely. Infinite lives.
(Although, nothing in life is indefinite or infinite. Subpixel shenanigans would make the setup fail over time. And no computer can count to infinity, so here we just mean "a lot" of lives.)
Discovering that exploit was a defining moment in video game history. It was glorious - the life counter would scroll past all known human digits, into the abstract and surreal. Yes, I do have left-triangle-upside-down-flag lives, thank you very much.
This is why we're called World 3-1. Because no matter if your first game was Mario or Minecraft, somewhere out there, if you just set things up right and test your boundaries, you too can achieve infinite lives.