Pokérus is a beneficial virus that affects one player at a time and lasts for 15 minutes.
You can "catch" Pokérus by interacting with the current host's Pokémon and Eggs. Every interaction gives you one "point." When the 15 minutes are up, everyone who interacted with the host has their points put into a draw, and the winner becomes the next host. This means that the more you interact, the higher your chances are, but there's never a guarantee!
Once you've had Pokérus, you become immune to it for one hour. After that, you will be able to participate and become the host again if you're lucky.
Note that any interactions so far in the day count, even if they were made before the recipient became the host, so it's okay if you already interacted with them today, your chances will be counted all the same.
In the unlikely event that nobody has interacted with the current Host, then it will automatically default back to a random Staff member and continue from there.
While under the effects of Pokérus, all interactions received give double EXP to your Pokémon and Eggs, and any interactions you make give you double Interaction Points. Powerful stuff!