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So, IVs are just basically a value that determines the power of your Pokemon's stats. They range in value from 0-31. 0 being the lowest 31 being the highest. The higher the number, the higher the stat. You won't see any immediate differences at first, but the full benefit comes at level 100 (everyone uses Wi-fi's lv 100 battle feature, so blah).
For example (Assuming no EVs):
A Jigglypuff with a 0 IV in HP at level 100 will have an HP stat of 340.
A Jigglypuff with an 8 IV in HP at level 100 will have an HP stat of 348.
A Jigglypuff with a 31 IV in HP at level 100 will have an HP stat of 371.
It's pretty basic for the most part, but those small differences could mean the difference between getting the KO or getting KOed. Unlike EVs, IVs are a constant value set when the Pokemon is caught, obtained as an egg, etc. They cannot be changed. They only way to get different IVs is to breed/catch another Pokemon.
A good calculator for checking IVs (assuming you also know what EVs your Pokemon has) is
Legendary Pokemon's Keep in mind that the most accurate results will only be obtained if the Pokemon is Level 100.
It's been a while. I cannot into coherent explanations.