>>22138734IVs are 6 invisible numbers between 0 and 31 that correspond to each stat, with 0 being absolute worst and 31 being perfect. At level 100(or 50? I forget), a Pokemon will have that amount of extra points in the related stat. Once a Pokemon's IVs have been determined (when the Pokemon is generated), they cannot be changed legitimately.
3 IVs are passed down to an egg by the parents when breeding. 5 will be passed down if a parent is holding a destiny knot. Luck, and not the holder of the destiny knot, determines which parent contributes which IVs. You can also pick a specific IV from a specific parent that you want to pass down with the "power" items (bracer, lens, etc.), but the Pokemon will still only pass down the regular amount, so destiny knot is more important for passing IVs, and since an everstone is needed to guarantee the nature of the child, there's really no room for power items on a breeding pair of Pokemon.
There is an IV checker in the Battle Resort Pokemon Center that will tell you a Pokemon's highest IV. The words you're looking for are "Can't be beat!". That's a perfect 31. He will list multiple stats if they are all the same.
