Pokérus is a rare but beneficial virus that your pokémon can catch. Read below to learn how to get Pokérus.
How to Get it
The easiest way to get Pokérus is to receive a Pokérus-infected pokémon in a trade, then spread it to your other pokémon (see How to Spread Pokérus below).
Otherwise, to get Pokérus, you can defeat or catch a wild pokémon who has it. To get Pokérus by fighting, it is not enough to just encounter a wild pokémon who has Pokérus and then run away; you have to actually defeat it. Unfortunately, there is no way to know if the wild pokémon has Pokérus. You just have to be lucky.
According to Bulbapedia, encountering Pokérus is even more rare than shiny pokémon. You have a 0.004069% chance of encountering Pokérus. In other words, one out of every 24,567 pokémon, statistically speaking, has Pokérus.
What it Does
Pokérus affects Effort Value (EV) growth. EVs are hidden bonus points that influence a pokémon's stat growth. When you defeat a pokémon, the pokémon who participated in the battle will all earn EVs based on which pokémon was defeated. For example, Rattata provides 1 Speed EV when defeated, so a pokémon who helps defeat a Rattata will get one hidden bonus point to its speed stat. A pokémon's EVs influence stat growth whenever the pokémon levels up.
A pokémon who has the Pokérus, or who has had it in the past, will get double the EVs from every pokémon it defeats. For example, defeating a Rattata usually gives you +1 speed EV, but a pokémon who has, or has had, Pokérus will gain +2 speed EVs after defeating a Rattata.
The Macho Brace is an item that does the same thing as Pokérus (doubling EVs earned) but the Macho Brace has the side effect of reducing the pokémon's speed in battle. Pokérus does not reduce the pokémon's speed in battle. However, a pokémon with Pokérus can wear the Macho Brace to get four times the EVs in battle.
Therefore, the overall effect of Pokérus is: pokémon who have had Pokérus will have larger stat increases when they level up.
How do I know if my Pokémon has Pokérus?
The only way to see if a particular pokémon of yours has the Pokérus is to look at its status summary. On the summary screen, next to its name, there will be a purple status badge that says PKRS, unless that pokémon has some other status, like sleep. However, having another status ailment at the same time as having Pokérus doesn't eliminate the Pokérus. It is still there, but you can't see it until the other status ailment has been cured.
This PKRS status badge will not be visible during battle. As mentioned previously, the Pokémon's status summary page is the ONLY place where you can see if that pokémon has Pokérus or not.
Also, if a pokémon in your party has Pokérus, then the next time that you have a nurse heal up your Pokémon, she will tell you that your pokémon might have Pokérus. Not long after that, you will get a phone call from the Pokémon Professor, telling you that he/she doesn't know what Pokérus does, but that it doesn't hurt the pokémon, so it's nothing to worry about.
The Pokérus infection goes away after a while, but the infected pokémon will permanently have the EV bonus, and there will be a small smiley face on that pokémon's status screen to show that it had Pokérus. Once it has the smiley face, it can no longer spread Pokérus to other pokémon.
How to Spread Pokérus
A pokémon who has Pokérus can spread it to other nearby pokémon. To do this, the pokémon with Pokérus must be in your active party. After you win a few battles, the pokémon that are next to the one with Pokérus will will also have Pokérus. Pokérus doesn't spread over time; you have to do battles in order to make it spread. Also, a pokémon in the PC box who has Pokérus can't spread it to other pokémon from there.
The order of pokémon in your active party goes in a zig-zag, often like this:
1 2 3 4 5 6
So, for example, if you have a pokémon with Pokérus in the #2 position, then the pokémon in #1 and #3 will have Pokérus after you complete a few battles. Similarly, for another example, if you have a Pokérus pokémon in position #4, then pokémon #3 and #5 will have Pokérus after a few battles.
To give Pokérus to friends, just give them a Pokérus-infected pokémon, or have them trade you a pokémon to infect. You might also be able to give them Pokérus if they defeat your Pokérus-infected pokémon in battle.
How Long it Lasts
A pokémon who is in your party will become immune to the Pokérus after approximately two days (but it can randomly be longer or shorter than that). The day is counted down at midnight. However, if you put a pokémon with Pokérus into a PC Box, then that pokémon's countdown won't decrease at all, which means that you can keep Pokérus indefinitely if at least one of your PC Box pokémon has it.
However, the EV-boosting effect of Pokérus lasts forever once a pokémon has had Pokérus. After the Pokémon becomes immune, the PKRS badge on the status screen will be replaced with a small smiley face to the right of that pokémon's picture, which represents the fact that that pokémon has had the Pokérus. Once this happens, that pokémon can no longer spread Pokérus to others, but it will always have the EV bonus from that point on.