Egg Moves are moves that a Pokémon can usually only learn via breeding. Or, they can learn it via level up or TM, but they can also get it from breeding as well for convenience.
(Those challenging the Little Cup, which uses base evolution Pokémon at level 5, will rely heavily on Egg Moves to get Little Cup Pokémon with formidable movesets.)
Both males and females can pass on Egg Moves, but since Egg Moves usually come from Pokémon of a different species, males are the most valuable for this, and females are used to chain breed if trying to get multiple Egg Moves on one offspring.
Example Using Rowlet
Let’s use an example: everybody’s favorite dapper owl, Rowlet!
Rowlet can learn Baton Pass, but via breeding only. This means we will have to have a female Rowlet breed with a father who knows the move. Togetic and Oricorio can learn Baton Pass by leveling up and are in the same Egg Group as Rowlet. Oricorio is the most easily accessible as a parent and learns Baton Pass at Level 16.
Any and all resulting eggs will hatch into a Rowlet guaranteed to have both the Nature that we chose and Baton Pass as one of its moves!
If Rowlet’s only potential parents for Baton Pass had to learn it through breeding as well, you simply repeat these steps to chain breed until you can finally put the move on your target Pokémon. It takes patience and research but is well worth your effort.
One particular Egg Group, however, is very, very blessed. Let’s talk about it and see how you can save hours if you are working within this Egg Group!