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An Egg move is any move passed down from a Pokémon breeding pair to their offspring, which it would not normally possess at the level it hatches. The term is also frequently used specifically for moves that a Pokémon can acquire only from breeding (i.e. moves that the offspring cannot learn via level-up, TM or HM, or Move Tutor), though a Pokémon can inherit moves from its parents in other ways.
Before breeding a Pokémon, one must consider whether to use any Egg moves in that Pokémon's moveset, and breed accordingly. Sometimes, to get a certain combination of moves onto a Pokémon (such as a Spinda with both Wish and Baton Pass) several steps must be taken; this is called chain breeding.
Prior to Generation VI, all methods of learning Egg moves required the father to know the move prior to breeding, which made certain combinations of moves impossible to possess; for example, Tentacruel can learn both Rapid Spin and Mirror Coat as Egg moves, but no Pokémon in the Water 3 Egg Group can possess both moves at the same time prior to Generation VI (since Tentacool and Tentacruel now can). (Such a combination is possible in Generation VI, if one parent knows Rapid Spin and the other knows Mirror Coat.)
Egg move breeding mechanics
If both parents know one or more moves that their offspring can learn by leveling up, the offspring will inherit that move. For example, Gengar and Mismagius both know Mean Look, so their child, Misdreavus (who learns the move at level 19) will know it when it hatches.
Note that Ditto cannot be used to pass down moves in this manner, because the only move it possesses is Transform.
TM and HM moves
Prior to Generation VI, if the male Pokémon knows a TM or HM move that the offspring can learn, the offspring will inherit that move. This applies whether or not the female Pokémon knows the move (or if Ditto was used in lieu of a female), and regardless of how the father originally learned the move.
In this example, the father, Gliscor, learns X-Scissor by level up, and his child, Joltik, can learn the same move through TM81, so it will know this move upon hatching.
Prior to Generation V, this was the only way to "reuse" one-of-a-kind TMs, like the ones obtained from Gym Leaders.
In Generation VI, the offspring no longer inherits TM or HM moves for this reason alone; however, some TM or HM moves can be inherited as Egg moves or due to being a level-up move. For example, Archen cannot inherit Roost (despite being compatible with TM19), but Fletchling can (because Fletchling learns it via level-up), and Vullaby can (because it is a specific Egg move for Vullaby).
If either parent (male Pokémon only prior to Generation VI) knows an Egg move of the baby's species, that child will be born knowing the move (similar to the inheritance of TM moves described above). In Generation VI this includes some, but not all, moves which are available in TM form.
For example, if a male Dragonite that knows Outrage is bred with a female Charizard that knows Flare Blitz, the resulting Charmander will know Outrage and Flare Blitz (only Outrage prior to Generation VI).
In addition to (or in the absence of) any inherited moves, the offspring Pokémon will know at least one default move when hatched, labeled in their level up move list as either "Start" or "--" in lieu of a level number. If this would result in the offspring possessing more than four moves, the inherited moves take priority and the moves on top of their level up move list (with "Start" or "--") will be the ones that are not learned. (If desired, the player may consult a Move Reminder to acquire those moves.)
Breeding with Incense
Incense-bred Pokémon often have a different or expanded Egg move list than their evolved forms, due to being classified in the Pokédex as a different species.
While not an Egg move, Pichu will know Volt Tackle if either of its parents holds a Light Ball.
Genderless Pokémon cannot inherit any Egg moves, as they can only be bred with Ditto, and always start with a fixed number of moves. Some of these, such as Beldum, are also incapable of learning moves via TM and HM.