Hidden Power is one of the most, if not the most, enigmatic moves in all of Pokémon. It can be learnt by just about any Pokémon, but not very many people know about its inner workings.

The game states that Hidden Power is a Normal-Type move with an undetermined power, 100% Accuracy, 15 Power Points (PP), and, in Diamond, Pearl, and Platinum, a Special Categorization (meaning that damage is calculated off of the Special Attack and Special Defense stats). While the Accuracy, PP, and Special Categorization are all set in stone, the Type and the Power, the two things not properly described in the In-game explanation, are unique to the Pokémon to which the move is taught. However, both of these factors can be determined by relatively simple formulas. Unfortunately, both of these formulas are based off of Individual Values (IVs), which are impossible to see in-game. From Pokémon X & Y, however, the power of the move became statically set at 60 with only the type varying.

There are several ways to determine the Type of the Hidden Power. Battling several wild Pokémon with different types allows you to figure out the Type through the process of elimination. To expedite the process if you are playing in Ruby, Sapphire, Emerald, or Diamond, you can battle a Kecleon to figure out the Type. Its Color Change Ability will inform you of the type unless it is Ghost, in which case the move will not affect the foe. Although the move's description lists it as being a Normal-Type move, this is the one type that it cannot be, so battling a Kecleon will work every time, assuming it does not KO you before you get to attack, or you KO the Kecleon with the move. There are, however, various people who tell you what your Hidden Power is in-game, such as the man in X & Y's Anistar City who gives you the TM. There is also a mathematical method to figure out the Type.

As I have mentioned before, IVs are used to calculate the Type and Power of each Pokémon's Hidden Power.
To determine the Type of the Hidden Power mathematically, one must use the following formula:

Type= ((H + A + D + S + X + Y)/ 4)mathfloor
The formula is a bit complex, but not too much so. The H, A, D, S, X, and Y variables correspond to the HP, Attack, Defense, Speed, Special Attack, Special Defense IVs respectively. If any particular IV is even, then its corresponding variable is 0. However, each variable gives a different number if the respective IV is odd. The H variable gives 1; the A variable gives 2; the D gives 4; the S, 8; the X, 16; and the Y, 32. Once the variables are determined, one then multiplies the sum by 15, and then divides it by 63. If the resulting quotient is a decimal, anything after the decimal is erased as a result of the "mathfloor" function.
Here's an example. Let's say we have a Pokémon whose IVs are as follows:
HP: 29
Attack: 19
Defense: 18
Speed: 28
Special Attack: 22
Special Defense: 15

Now, since the HP IV is odd, it goes into the formula as H=1.
Type= ((1 + A + D + S + X + Y)/ 4)mathfloor

And since the Attack IV is odd, it goes into the formula as A=2.
Type= ((1 + 2 + D + S + X + Y)/ 4)mathfloor

Since I think you get the process, let's finish plugging in the numbers:
Type= ((1 + 2 + 0 + 0 + 0 + 32)/ 4)mathfloor

Now add up the former variables:
Type= ((35)/ 4)mathfloor

Multiply 35 by 15:
Type= (525/63)mathfloor

And divide 525 by 63:
Type= (8.333)mathfloor

And, since there are numbers after the decimal, we eliminate them due to "mathfloor."
Type= 8

Wait, there is no type called eight! Well, there is a table that tells us which numbers correspond to which Hidden Power Type.
0 = Fighting
1 = Flying
2 = Poison
3 = Ground
4 = Rock
5 = Bug
6 = Ghost
7 = Steel
8 = Fire
9 = Water
10 = Grass
11 = Electric
12 = Psychic
13 = Ice
14 = Dragon
15 = Dark

So this Pokémon has a Hidden Power Type of Fire because the equation produced an "8."

Even though we now know that this Pokémon has a Hidden Power of the Fire Type, we still don't know its Power, which is certainly important! Unfortunately, there is no really easy way to figure it out, so mathematics is required.

Power: ((H + A + D + S + X + Y)*(40/63) + 30)mathfloor

It looks fairly similar to the Type formula, doesn't it? However, it doesn't use even and odd numbers to change the variables' values. Instead, it asks if the IV, when divided by 4, has a remainder of 2 or 3 (in decimal, if it has .5 or.75 after division). If the any of the IVs, when divided by 4 do not have a remainder of 2 or 3, the corresponding variable has a value of 0. If the IV for the respective stat is divisible by 2 or 3, the values of the variables are as follows: H is equal to 1; A is equal to 2; D, equal to 4; S, to 8; X, to 16; and Y, to 32. Again, once the values of the variables are determined, add them up. Now, multiply them by 40, divide the product by 63, and then add 30 to the result. Again, you will want to use the "mathfloor" function to eliminate any numbers after the decimal place (if there is one).

Now that the method for determining the Power has been established, let's figure out the strength of the example Pokémon's Hidden Power.
To recapitulate, here are that Pokémon's IVs:
HP: 29
Attack: 19
Defense: 18
Speed: 28
Special Attack: 22
Special Defense: 15

Now, 29 divided by 4 is 7 with a remainder of 1. Since that's not 2 or 3, the H variable is 0.
Power: ((0 + A + D + S + X + Y)*(40/63) + 30)mathfloor

Moving on to the Attack IV, 16 divided by 4 is 4 with a remainder of 3.
Power: ((0 + 2 + D + S + X + Y)*(40/63) + 30)mathfloor

For all of the others: 18 divided by 4 is 4 with a remainder of 2; 28 divided by 4 is 7 with a remainder of 0; 22 divided by 4 is 5 with a remainder of 2; and 15 divided by 4 is 3 with a remainder of 3. So, D, S, X, and Y are 4, 0, 16 and 32 respectively.
Power: ((0 + 2 + 4 + 0 + 16 + 32)*(40/63) + 30)mathfloor

Next, add up the numbers:
Power: ((54)*(40/63) + 30)mathfloor

Multiply 54 by 40:
((2160/63) + 30)mathfloor

Divide 2160 by 63:
(34.2857 + 30)mathfloor

And add 30:
(54.2857)mathfloor

Again, since there are decimals, "mathfloor" eliminates them, so this Pokémon has:
A Fire-Type Hidden Power with a Base Power of 54. Not too bad.

A few things to note about Hidden Power:
-The most important IV to control when aiming for a good Hidden Power is Special Defense. Without a Special Defense IV that produces a remainder of 2 or 3 when divided by 4, the highest power possible is 49.
-Prior to XY, most of the time, one will want a Hidden Power with a Power above 60. However, the Ability Technician will benefit Pokémon with Hidden Power of 60 or below, because it multiplies the Base Power of moves with a Base Power of 60 or lower by 1.5.
-Hidden Power's strange mechanics allow for a great variance in IVs to produce the same Power and Type. A Pokémon with IVs of all 3s has the same Power and Type as one with all 31 IVs.
-There is no Fairy-type Hidden Power

Thanks to Dragonair for writing this for us