Individual Values - The Basics
Individual Values (IVs for short) are the equivalent of Pokémon genes, directly affecting what each individual Pokémon is good at. Unlike Effort Values, IVs are not directly manipulable by the player.
For a real life analogy, imagine two basketball players. One has a natural talent for perimeter defense, while another is naturally very speedy and is proficient at starting fast breaks.
Pokémon are similar. Every single one has a unique set of IVs that determine which of their stats are naturally higher or lower than the average. Given two Pokémon of the same species, for example, the one with higher Speed IVs is naturally faster than the other one and thus will almost always have a higher speed stat.
For a real life analogy, imagine two basketball players. One has a natural talent for perimeter defense, while another is naturally very speedy and is proficient at starting fast breaks.
Pokémon are similar. Every single one has a unique set of IVs that determine which of their stats are naturally higher or lower than the average. Given two Pokémon of the same species, for example, the one with higher Speed IVs is naturally faster than the other one and thus will almost always have a higher speed stat.
How IVs Affect Pokémon
Each Pokémon has 6 different IVs that correspond to its 6 stats: HP, Attack, Defense, Special Attack, Special Defense, and Speed. The IVs are a random number from 0-31, with 0 being lowest possible value and offering no stat bonuses at all, and 31 being the maximum value. The Pokémon's IV has a one-to-one ratio with its stats: having 31 IVs in Speed, for example, gives the Pokémon an immediate boost of 31 to the Speed stat at level 100.
Manipulating IVs
IVs are determined at random when you encounter a Pokémon in the wild or hatch one from an egg. Thus, you cannot directly manipulate all of your Pokémon's IVs.
The only way to indirectly manipulate IVs for your Pokémon is through Breeding(LINK). From Pokémon Platinum onwards, the breeding process was coded in a way that the offspring would always inherit 3 random different IVs from its parents, with the other 3 IVs being determined at random. Giving a parent one of the "Power" items would let you choose which IV would be inherited by the offspring.
The only way to indirectly manipulate IVs for your Pokémon is through Breeding(LINK). From Pokémon Platinum onwards, the breeding process was coded in a way that the offspring would always inherit 3 random different IVs from its parents, with the other 3 IVs being determined at random. Giving a parent one of the "Power" items would let you choose which IV would be inherited by the offspring.
In the above example, the newly hatched Charmander inherited 3 random IVs: Its father's Special Attack IV of 31, and its mother's 31 Defense IVs and 30 Speed IVs. The rest of its IVs are determined at random.
Further Reading
Click HERE for a more in-depth explanation of Individual Values on Bulbapedia, covering the math behind how it affects stats and the history of Individual Values throughout the generations of Pokémon.