Since the second generation of Pokémon games, Pokémon have been able to reproduce to create new Pokémon. Pokémon can learn new moves through breeding, and furthermore some Pokémon are only obtainable with this method.
Breeding occurs at the Pokémon Daycare. If two compatible Pokémon are left with the daycare lady, they will produce an egg. (The games are purposely vague about how this happens.) Pokémon are compatible for breeding if:
They are not legendary Pokémon, baby Pokémon, Unown, Nidorina or Nidoqueen. They are of opposite genders. They are in the same egg group (see table).
A Pokémon meeting the first criteria can also breed with Ditto. Genderless Pokémon can only breed with Ditto.
Once you take the egg and walk around with it for a while, it will hatch into a level 1 Pokémon (or level 5 in Generations 2-3). The Pokémon that hatches will be the same species as the female, but at the bottom of the evolutionary chain. For example, breeding a female Blastoise with a compatible Pokémon would generate a Squirtle egg. In the case of Ditto, the Pokémon egg will always be the non-Ditto Pokémon - so if you breed Ditto with a male Charizard, the egg would be a Charmander.
There are two sets of male-female counterparts: Nidoran♀/Nidoran♂ and Illumise/Volbeat. Breeding one of these with a compatible Pokémon will give you either the male or female variant. For example, breeding a male Golduck with Nidoran♀ would yield either Nidoran♀ or Nidoran♂.
Note that in Generation 2-4 breeding the male variant with Ditto only produces a male egg.
Some Pokémon also produce variable eggs, based on the item held by the parent. When a particular Incense is held, the offspring will be a baby Pokémon, otherwise it will be the next stage up in the evolutionary line. For example, breeding a female Roserade will produce a Roselia egg, but if the Roserade is holding a Rose Incense, it will produce a Budew egg. See the list of baby Pokémon below for details.
The Pokémon Manaphy and Phione are listed in the Water 1 and Fairy egg groups, however they cannot breed with others in that group, only with Ditto. For each of them the result is a Phione egg, but Phione does not evolve into Manaphy.
The main purpose of breeding is usually to obtain a Pokémon that knows certain moves. If the male Pokémon knows moves that the baby Pokémon is capable of learning, the baby will know them when it hatches from the egg. Prior to Generation 5 this was a good way to reuse TMs that you taught a Pokémon and cannot re-obtain (TMs are no longer passed down by breeding).
The baby will know any move that it learns at level 1. If both parents know a move that the child would learn by level up, the child will also know it upon hatching. For example, breeding two Ampharos knowing Thunder will deliver a Mareep knowing Thunder at level 1.
Furthermore, there are some moves Pokémon can only learn by breeding - these are called egg moves, and are listed in our Pokédex alongside the other moves. In most cases, another Pokémon in the same egg group learns the move by level up and can pass it on by breeding, but sometimes you need to chain breed from one Pokémon to another to another.
If there are too many candidate moves that the baby can learn, they follow this precedent, with each new move overwriting previous ones:
A female Pokémon (or male when bred with Ditto) has a 60% chance of passing down its ability to the offspring; otherwise the offspring's ability is randomly chosen from its regular abilities. In practice, this means that if the parent has a regular ability, there is an 80% chance the child has the same ability slot (60 + 20) and 20% chance it has the other slot. For hidden abilities, the child has a 60% chance of having its hidden ability and 20% chance for each of its regular abilities (or 40% if it only has one regular ability).
Note that the ability itself may be different if the Pokémon evolutions have different ability options. For example, Poochyena has the abilities Run Away or Quick Feet, while Mightyena has Intimidate or Quick Feet. A Poochyena bred from a Mightyena with Intimidate has an 80% chance of having Run Away and 20% chance of Quick Feet. If the Mightyena has Quick Feet, Poochyena has an 80% chance of Quick Feet and 20% chance of Run Away.
Note that in Gen 3-4, the ability of the hatched egg is randomly chosen from the two possibilities (50% each) regardless of the parents' abilities. In Gen 5, only females could pass down abilities; males bred with Ditto reverted to the 50/50 chance for each ability.
Pokémon can also inherit natures while breeding. Normally the nature is chosen at random from the 25 possibilities, but as natures raise and lower certain stats, often you will want a specific nature. So if a Pokémon is holding the Everstone item, its child is guaranteed to have the same nature. (If both parents hold an Everstone the nature is chosen at random between the two.)
Note that prior to Pokémon Black 2/White 2, natures only have a 50% chance of being passed down. In Emerald it only applies to the female parent or Ditto. And in Gen 4 both parents must be from the same language game.
From Generation 3 (Ruby/Sapphire) onward, Pokémon offspring will also inherit some of the Individual Values from the parents. (IVs are hidden values that improve your final stats.)
The baby will inherit three stats from either parent. So if you were breeding a male Infernape and female Ninetales, the resulting Vulpix could inherit the HP and Defense IVs from Infernape and the Speed IV from Ninetales.
The IVs chosen are random and in the case of the same one being selected twice, the former would be overwritten with the latter. For example, if the game chose Ninetales' HP and Attack, then Infernape's Attack, then the baby would only inherit two IVs - the HP from Ninetales and Attack from Infernape.
From HeartGold/SoulSilver onward a new mechanic was added to control this. If either parent is holding one of the EV-training Power items then the child will inherit the corresponding IV. Two different IVs are randomly inherited from the parents as normal, and the remaining three are completely random. If both parents hold a Power item then one of the two stats is chosen at random. The power items are:
So for example, holding the Power Belt means the baby will inherit the Pokémon's Defense IV and two random non-Defense IVs from either parent.
From X/Y onward, a parent holding the Destiny Knot item causes the baby to inherit five IVs from that parent. If the other parent holds a Power item, the baby will inherit the corresponding IV and then four random IVs from the Destiny Knot parent.
Some Pokémon may only be obtained by breeding - these are known as baby Pokémon. Many baby Pokémon can be obtained in the wild in later games. Several baby Pokémon also require an item to be held by the parent.
