Egg groups determine which Pokémon are able to interbreed in Pokémon breeding. Pokémon were first able to breed in the Second Generation games Pokémon Gold and Pokémon Silver. Breeding allowed Pokémon to pass down DVs and moves to their children. Moves usually are inherited from the male Pokémon, while the child takes on the form of the mother in her basic form. A baby would hatch out of an egg after a specified number of steps in the game and would always be at level five.

When breeding, one typically has a 50/50 chance of getting male or female Pokémon. If the Pokémon is a Starter Pokémon, or any of a number of other types which have different gender ratios than 50-50, the results are skewed. Some Pokémon are genderless; some species can be only male or only female.

Breeding Chains

Many Pokémon are included in two different egg groups and can form what are known as breeding chains. These link many unrelated Pokémon together, allowing moves that would not normally be accessible to certain Pokémon to be legal, like Heal Bell on Blissey.

TMs are also passed down through breeding through male Pokémon. Doing this would cause the moves to appear in the Pokémon's moveset if said Pokémon can learn the move via TM or Egg Move.

Egg Groups

Ditto

Ditto can breed with any Pokémon except for Legendary Pokémon. Ditto can also breed with Pokémon such as: Baltoy, Voltorb, Porygon and other genderless Pokémon.

No eggs

The following Pokémon are genderless and therefore incapable of breeding, with some exceptions made for Ditto. Those exceptions are the nonlegendary Pokémon that are not babies. The list is as follows:

Shedinja, Azurill, Magnemite, Magneton, Voltorb, Electrode, Lunatone, Solrock, Baltoy, Claydol, Igglybuff, Staryu, Starmie, Pichu, Wynaut, Beldum, Metang, Metagross, Regirock, Regiice, Registeel, Latias, Latios, Kyogre, Groudon, Rayquaza, Jirachi, Deoxys, Nidorina, Nidoqueen, Articuno, Zapdos, Moltres, Mewtwo, Mew, Cleffa, Togepi, Unown, Tyrogue, Smoochum, Elekid, Magby, Raikou, Entei, Suicune, Lugia, Ho-Oh, Celebi. (Group X)

Plant

Many Grass-type Pokémon belong to this group:

Lotad, Lombre, Ludicolo, Seedot, Nuzleaf, Shiftry, Shroomish, Breloom, Oddish, Gloom, Vileplume, Bellossom, Cacnea, Cacturne, Tropius, Bulbasaur, Ivysaur, Venusaur, Paras, Parasect, Bellsprout, Weepingbell, Victreebel, Exeggcute, Exeggcutor, Tangela, Chikorita, Bayleef, Meganium, Hoppip, Skiploom, Jumpluff, Sunkern, Sunflora. (Group 1)

Bug

Many Bug-type Pokémon belong in this group:

Wurmple, Silcoon, Beautifly, Cascoon, Dustox, Surskit, Masquerain, Nincada, Ninjask, Volbeat, Illumise, Trapinch, Vibrava, Flygon, Pinsir, Heracross, Caterpie, Metapod, Butterfree, Weedle, Kakuna, Beedrill, Paras, Parasect, Venonat, Venomoth, Scyther, Ledyba, Ledian, Spinarak, Ariados, Yamna, Pineco, Forretress, Gligar, Scizor, Shuckle.(Group 2)

Flying

This is the group to which most of the Flying-type Pokémon belong:

Taillow, Swellow, Wingull, Pellipper, Zubat, Golbat, Crobat, Doduo, Dodrio, Skarmory, Swablu, Altaria, Natu, Xatu, Pidgey, Pidgeotto, Pidgeot, Spearow, Fearow, Farfetch'd, Aerodactyl, Hoothoot, Noctowl, Togetic, Murkrow.(Group 3)

Humanshape

The humanoid Pokémon belong here, including many Fighting-type and Psychic-type Pokémon:

Abra, Kadabra, Alakazam, Makuhita, Hariyama, Sableye, Machop, Machoke, Machamp, Meditite, Medicham, Volbeat, Illumise, Spinda, Cacnea, Cacturne, Drowzee, Hypno, Hitmonlee, Hitmonchan, Hitmontop, Chansey, Blissey, Mr. Mime, Jynx, Electabuzz, Magmar.(Group 4)

Mineral

Rock-type Pokémon, Ground-type Pokémon and a Steel-type Pokémon belong in this group.

Geodude, Graveler, Golem, Nosepass, Onix, Steelix, Snorunt, Glalie, Porygon, Porygon 2, Sudowoodo.(Group 5)

Indeterminate

Poison-type Pokémon, Ghost-type Pokémon and a few others beclong in this group.

Ralts, Kirila, Gardevoir, Gulpin, Swalot, Slugma, Magcargo, Grimer, Muk, Koffing, Weezing, Castform, Shuppet, Banette, Duskull, Dusclops, Chimecho, Wobbuffett, Gastly, Haunter, Gengar, Misdreavus.(Group 6)

Ground

Fire-type, Water-type, Normal-type, Dark-type, Electric-type and Ice-type Pokémon belong to this group.

Torchic, Combusken, Blaziken, Poochyena, Mightyena, Zigzagoon, Linoone, Seedot, Nuzleaf, Shiftry, Slakoth, Vigoroth, Slaking, Whismur, Loudred, Exploud, Skitty, Delcatty, Mawile, Electrike, Manectric, Wailmer, Wailord, Numel, Camerupt, Torkoal, Spoink, Sandshrew, Sandslash, Spinda, Zangoose, Seviper, Kecleon, Absol, Vulpix, Ninetales, Pikachu, Raichu, Psyduck, Golduck, Girafarig, Phanpy, Donphan, Rhyhorn, Rhydon, Spheal, Sealeo, Walrein, Rattata, Raticate, Ekans, Arbok, Nidoran female, Nidoran male, Nidorino, Nidoking, Diglett, Dugtrio, Growlithe, Arcanine, Ponyta, Rapidash, Farfetch'd, Seel, Dewgong, Tauros, Eevee, Vaporeon, Jolteon, Flareon, Espeon, Umbreon, Cyndaquil, Quilava, Typhlosion, Sentret, Furret, Mareep, Flaaffy, Ampharos, Aipom, Wooper, Quagsire, Dunsparce, Snubbull, Granbull, Sneasel, Teddiursa, Ursaring, Swinub, Piloswine, Delibird, Houndour, Houndoom, Smeargle, Milktank.(Group 7)

Water 1

This is the group to which most of the Water-type Pokémon belong:

Mudkip, Marshtomp, Swampert, Lotad, Lombre, Ludicolo, Wingull, Pellipper, Surskit, Masquerain, Goldeen, Seaking, Marill, Azumarill, Corphish, Crawdaunt, Feebas, Milotic, Psyduck, Golduck, Spheal, Sealeo, Walrein, Clamperl, Huntail, Gorebyss, Corsola, Horsea, Seadra, Kingdra, Squirtle, Wartortle, Blastoise, Poliwag, Poliwhirl, Poliwrath, Politoed, Slowpoke, Slowbro, Seel, Dewgong, Lapras, Omanyte, Omastar, Kabuto, Kabutops, Dratini, Dragonair, Dragonite, Totodile, Croconaw, Feraligatr, Wooper, Quagsire, Remoraid, Octillery, Delibird, Mantine.(Group 8)

Water 2

This is the group to which some Water-type Pokémon belong:

Goldeen, Seaking, Magikarp, Gyarados, Carvanha, Sharpedo, Wailmer, Wailord, Barboach, Whishcash, Relicanth, Chinchou, Lantern, Luvdisc, Shellder, Cloyster, Quilfish, Remoraid, Octillery.(Group 9)

Water 3

This is the group to which some Water-type Pokémon belong:

Tentacool, Tentacruel, Corphish, Crawdaunt, Lileep, Cradily, Anorith, Armaldo, Corsola, Omanyte, Omastar, Kabuto, Kabutops, Remoraid, Octillery.(Group 10)

Monster

These are Pokémon that are monster-like.

Treecko, Grovyle, Sceptile, Mudkip, Marshstomp, Swampert, Whismur, Loudred, Exploud, Aron, Lairon, Aggron, Bulbasaur, Ivysaur, Venusaur, Charmander, Charmeleon, Charizard, Squirtle, Wartortle, Blastoise, Nidoran♀, Nidoran♂, Nidorino, Nidoking, Slowpoke, Slowbro, Slowking, Cubone, Marowak, Likitung, Kangaskhan, Lapras, Snorlax, Chikorita, Bayleef, Meganium, Totodile, Croconaw, Feraligatr, Mareep, Flaaffy, Ampharos, Larvitar, Pupitar, Tyranitar.(Group 11)

Fairy

These Pokémon are considered to be shaped like fairies, and as such are part of the Fairy breeding group. They are:

Shroomish, Breloom, Marill, Azumarill, Skitty, Delcatty, Mawile, Plusle, Minun, Roselia, Jigglypuff, Wigglytuff, Castform, Pikachu, Raichu, Snorunt, Glalie, Clefairy, Clefable, Hoppip, Skiploom, Jumpluff, Snubull, Granbull.(Group 12)

Dragon

Dragon-type Pokémon, along with those that resemble Dragons, are in this group.

Treecko, Grovyle, Sceptile, Magikarp, Gyarados, Swablu, Altaria, Seviper, Feebas, Milotic, Horsea, Seadra, Kingdra, Bagon, Shelgon, Salamence, Charmander, Charmeleon, Charizard, Ekans, Arbok, Dratini, Dragonair, Dragonite.(Group 13)

Special Notes

Other important facts about breeding:
Azurill and Wynaut can be bred only by giving the father(/mother?) a Sea or Lax Incense to hold. (respectively)
Pichu will acquire Volt Tackle if the father(/mother?) is holding a Light Ball, but only in Pokémon Emerald.
Also only in Emerald, if either (?) parent is holding an Everstone, then the nature of that parent will be passed down.