Egg Moves are moves most passed down from a Pokémon breeding pair to their offspring. This can happen in one of four different ways.
If both parents know one or more moves that their offspring could learn by leveling up, the offspring will have that move.
If the male Pokémon knows a TM or HM move that the offspring can learn, the offspring will have that move. For example, if a male Dragonite knowing Outrage is bred with a female Charizard, the resulting Charmander will know Outrage.
All offspring Pokémon, in the absence of passed-down moves, will know at least one move when hatched, labeled in their level up move list as either "Start" or "--" of a level number. If these moves, together with the other egg moves obtained in the above ways, number more than four, then the moves on top of their level up move list (with "Start" or "--") will be the ones that do not show up.
Most online Pokédex list moves from the third category as a Pokémon's Egg Moves. Before breeding a Pokémon, one must consider whether or not to use any egg moves in that Pokémon's moveset, and breed accordingly. To learn more than one egg move, since they are only passed on by the father, only a father Pokémon in the appropriate Egg Group that can legitimately have all of the moves in one moveset can be used. This makes certain combinations of egg moves impossible for certain Pokémon. An example would include something like Tentacruel having both Rapid Spin and Mirror Coat as available egg moves but no Pokémon in the Water 3 Egg Group can learn both moves so Tentacruel can't use both in the same moveset.
Sometimes to get a certain moveset onto a Pokémon, such as a Gible with Outrage and Thrash several steps must be taken. This is called chain breeding.
Exceptions Certain Pokémon, like Magikarp and Beldum, cannot inherit any egg moves and start with only Splash and Take Down, respectively.
When Mudkip is breeded when Swampert knows waterfall and surf, Mudkip will have these moves along with the other moves when it starts at its first level.
Pichu can "inherit" Volt Tackle if the female Pikachu or Raichu of the breeding pair holds the Light Ball.
Breeding for Egg Moves
If a person wishes to put in a certain amount of effort into breeding it is possible to breed a Tyrogue inheriting moves from each of its evolved forms (Hitmonlee, Hitmonchan, & Hitmontop) by carefully breeding of several Tyrogue's. One starts by either evolving a Tyrogue into one of if evolutions (unless you already have one of its evolutions use it) and then have it learn any of that form's moves that Tyrogue can inherit as an Egg Move (for example if it a Hitmonlee one can teach it Hi Jump Kick) then breed it with a Ditto to produce another Tyrogue and have it evolve into a different form from the one it was breed from. Via a careful process of Breeding, Evolution, and Breeding it is possible to breed a Tyrogue with move set consisting entirely of Egg Moves from each of its evolutions. An example of a possible moveset produced via this method could be Hi Jump Kick (Hitmonlee), Vacuum Wave (Hitmonchan), Rapid Spin (Hitmontop), Bullet Punch (Hitmonchan).
Cross-breeding Egg Moves
An advanced version of this method allows one to obtain certain Egg Moves from a Pokémon of a different egg group. For example it is possible to create a Lickitung with the egg move Belly Drum by first breeding a male Poliwhirl (that knows Belly Drum) with a female Slowpoke to produce a male Slowpoke, which will inherit Belly Drum as an egg move and belongs to both Water 1 Egg Group and Monster Egg Group (basically bridging the gap between Poliwhirl's Water 1 group & Lickitung's Monster group). Then carefully breed the male Slowpoke with a female Lickitung, producing a Lickitung with egg move, Belly Drum (this is just one example of obtaining Egg Moves via carefully planned cross-breeding).