Like real-life viruses, Pokerus can spread amongst your Pokemon. More specifically, after a battle (or escaping a battle), Pokerus can spread between adjacent Pokemon in your battle party, provided they aren't immune to Pokerus.
Furthermore, Pokemon will automatically become cured of Pokerus over time. When this occurs, a little smiley face icon will appear below their picture. Cured Pokemon will become immune to Pokerus but will retain all the benefits of Pokerus.
Once you have Pokerus, it's a good idea to try and contain it, so you spread it at will and reap its benefits. Pokemon with Pokerus can be stored inside a PC, preventing the virus from spreading or disappearing. Later, you can withdraw your Pokemon and gleefully spread the virus.