It's (Sort of) a Snake!
Paper.io is a weird online multiplayer game of sort of Snake, where you have to capture more and more territory by directing your constantly-moving square in order to trace out new chunks of whatever your chosen color is. The catch is that if another player runs over a line you're tracing (before it's completed), you're toast. So it's sort of like Snake, except running into other player's lines is a good thing.
The be all and end all of Paper.io is expansion. Simple in theory, but made much more difficult in practice thanks to other players who are likely just as driven (and blood-thirsty) as you are.
- Start small. As soon as you spawn you'll be in motion, which can lead to immediate elimination if you aren't watching where you're going. Try to keep your first couple of paths small in order to give yourself time to get a feel for the area around you, and give surrounding players less opportunities to take you out.
- Watch your neighbors. It won't take long for you to start butting heads with other players. Try to keep your expansion to sides that only have one or two potential threats and don't make any big moves if you can see their square – because they'll likely move in for the kill as soon as you give them an opening.
- If you see another player's square heading in your direction, close up your path quick! They can't take you out if you don't give them the chance, and chasing you down could leave them vulnerable as well.
- Know your own limits. Things like running into your own path – as opposed to your more solid mass – and simply moving backwards will cause you to eliminate yourself (too easily, I think) if you aren't careful.
- Collect gems. Every time a player is take out, they'll leave a few of these gem-looking things behind. Collect them to give your score a little boost.
- You can change your controls in the options menu. If the swiping doesn't work for you, tap the gear icon on the title screen and then the finger icon. You can pick from swipe, virtual buttons on the left, or virtual buttons on the right. I'd recommend sticking with the swiping, honestly.
All of the .io games are essentially a simple game made more complicated by turning it into a multiplayer free-for-all. It's not a bad concept, really, but I feel it may have peaked with Slither.io, which was basically the game Snake with fewer movement constraints but also on a Massively Multiplayer scale. Paper.io is sort of like that, only it's not as fun.