What is it?
PokeRus is a very rare "virus" that spreads amongst party Pokemon. Once a Pokemon has PokeRus, it gains double of the EVs it would normally gain from Pokemon battling.
For example, a normal Pokemon would gain 1 Speed EV when defeating a Magikarp.
A Pokemon with PokeRus would gain 2 Speed EVs when defeating a Magikarp.
So, you obviously would want to spread this! It spreads after battling with your party Pokemon. There is no definite number of battles it takes to spread, but it does spread rather quickly.
More technical tidbits here.
Does it disappear?
After a certain amount of "days" (internal clock passing midnight) a Pokemon is cured of PokeRus and cannot spread it any longer. However, it still retains the positive effects of it, meaning it still gains double EVs! A Pokemon that is cured of PokeRus has a icon on its status/summary screen.
You can keep it from disappearing and always have a Pokemon that can spread it if you keep a Pokemon with PokeRus in your PC.
Horray! You've got PokeRus!