1. The condition is called "Pokerus," and it is good. Very, very good. And extremely rare.
2. Deposit at least one of the Pokemon in your team that still has the virus active into one of the PC boxes (it doesn't matter which) immediately. If you do not, the beneficial virus will leave in somewhere around a day - whether the game is turned on or off.
3. Okay, now that that's out of the way... do you know anything about EVs? If not, they're basically points that Pokemon accumulate in each stat which increase their abilities in those respective stats. Sort of like experience points, but with each individual stat. I'm not going to take the time to explain them here; you can find that info in other threads.
4. Did I mention that you should deposit one of your infected Pokemon in the PC? If you do that, it will effectively "freeze" the virus, thus preventing it from wearing off for that particular Pokemon. I'd recommend depositing more than just one if you can.
5. Once again, you are very, very lucky to have caught this virus. So make sure you don't lose it! Make haste and deposit!
EDIT: If you want more information about Pokerus, you can find it in this thread. For now, I'd recommend storing a Pokemon to make sure you don't lose the virus, then reading up on all the details.