IV's are something the people that actually care to win every single battle use to their advantage. Essentially, even though they may be of the same species, virtually every Pokemon you catch is unique in a small sense. IV's are the means to allow this. Basically, every single stat (HP, Attack, Defense, etc.) has a random value from 0-31 assigned to it upon capture. For every value above "0" by the time you reach level 100, a Pokemon's total stats will be raised by twice the value of that number. For example, if your Pikachu had a special attack IV of 15, by the time that Pikachu reaches level 100, it's special attack will be 30 points higher than the lowest possible special attack for Pikachus, and 32 points lower than the highest obtainable special attack stat, barring the effects of Natures and EVs. I think they are called individual values.
The Nature your Pokemon has (Lonely, Mild, etc.) Can add 10% to one stat, while subtract 10% from another. This can be very useful apparently, but let me state that there aren't Natures to raise or lower the HP stat, unlike IVs and Evs.
EVs are actually effort values. Essentially, this is the thing Pokerus doubles the amount you receive, and is probably the reason why you may have noticed irregular stat boosts upon a few level-ups. Different Pokemon you battle give hidden points, and upon accumulation of enough of these points in one stat, you gain a one-point in the respective stat. There is a limit of 510 of these points on a Pokemon, and the points can actually be removed through new berries. To use these stats to the maximum effectiveness, you would have to train for some time against specific pokemon. Also, there are new items created to make this training easier.
Now, most players choose to ignore much of these factors, as paying attention to them causes immense attention to the game, beyond realistic acceptibilities. However, as time passes, online play will become more and more competitive between certain groups by virtue of taking advantage of these stat rules. I assure you that I have not ever taken part in any of this kind of stuff, and it is not the reason why I dominate you in a Pokebattle, though at some point in time, perhaps a year or so from now, it may be.