Individual Values: Everyone is Unique

You may have been battling in the Battle Tower with, for example, your Golbat, and the enemy may happen to be using a Golbat as well. You are the same level, an you attack with Bite. It does decent damage, let's say it takes a quarter of your health away. The enemy Golbat then uses Bite on you as well. You lose half of your health. What happened here? "Individual Values" happened.

Individual Values, or IVs for short, determine how powerful a pokemon will be, and they cannot be changed. All pokemon have 6 IVs, one for each stat, and each IV ranges from 0-31. Once they are hatched from the egg, the IVs are set in stone. This is stepping into something I am not fully knowledgable on, and many websites make things vague or too much to swallow. I will tell what I know to be true, and please contribute if you know something I have not listed.

Let's start with an example to show you how much IVs can affect a pokemon. This example is taken from "gamespot.com";

For this set, a Hardy Heracross has an IV of 0 in every stat.

HP: 270
Atk: 255
Def: 155
Spa: 85
SpDef: 195
Spe: 175

Now to show you what a Hardy Heracross looks like with every IV at 31.

HP: 301
Atk: 286
Def: 186
Spa: 116
SpDef: 226
Spe: 206

You can see the difference in the stats of those Heracross. That's due to their difference in IVs. Natures play a role in IVs, but as you can see from the example, there's more to it than just that, since those Heracross had the same Nature.

Breeding for IVs;

The parents of an egg pass down IVs to their children. These children have the potential to take these IVs and improve them in their own IVs. So if you breed the children and then breed their children, and continuously breed children, getting better and better IVs each time.
NOTE: When passing down IVs, you should know that each parent passes down 1 IV, and then one of the parents will pass down a second IV. The remaining 3 IVs the child will have are chosen at random, and are not dependent upon a parent. In HG/SS and beyond, if a parent is holding one of the "Power Items" that can be purchased in the Frontier/Subway, the IV that that item represents will be passed down to the child. If both parents are holding a Power Item, only one of those stats is garenteed to pass down.

So that is how to alter IVs in the egg. It will prove to be very useful. You can find out your pokemon's best IV by talking to the man in the Battle Tower in Generation IV, or the Subway in Generation V. The Stat Judge, as he is called, will tell you a pokemon's "overall potential" and its highest IV stat.

"Overall Potential" is the total amount of IVs a pokemon has, ranging from 0-186. But he will not give you a number, but rather a ranking, and each ranking has its range of IVs. They are as follows;

"Decent": 0-90
"Above Average": 91-120
"Relatively Superior": 121-150
"Outstanding": 151-186

The Stat Judge will do the same for your highest IV stat. He'll tell you a stat, then a description of it, and these descriptions also represent values. They are as follows;

"Rather Decent": 0-15
"Very Good": 16-25
"Fantastic": 26-30
"Can't Be Better": 31(perfection!)

The characteristic on your Pokemon's description page is a representation of your Pokemon's highest IV. The chart above tells you what each characteristic indicates. I found the information from  http://bulbapedia.bulbagarden.net/wiki/Characteristic and made my own chart because I couldn't just copy-paste their's.

Brock out.