An Egg Group is one of 15 breeding compatibility categories, introduced in Generation II, that decides which Pokémon can breed with each other. A Pokémon belongs to one or two Egg Groups, and Pokémon in the No Eggs Discovered group cannot breed at all.
The 15 groups are Monster, Water 1, Water 2, Water 3, Bug, Flying, Field, Fairy, Grass, Human-Like, Mineral, Amorphous, Dragon, Ditto, and No Eggs Discovered. Two Pokémon can produce an egg if they share at least one group and are of opposite genders, with Ditto as the universal exception: a Ditto breeds with anything outside the No Eggs Discovered group, regardless of gender. Field is the largest group, which is why so many routes to a good egg move chain run through a Field Pokémon.
Dual group Pokémon are the bridges of breeding. A male from one group passes an egg move to a female of another group, and their child, if it belongs to both, can carry that move onward. Competitive players plan these chains on paper before hatching anything. Smeargle, a Field group Pokémon that can learn nearly any move with Sketch, is the classic shortcut for passing moves within its group.
No Eggs Discovered covers baby Pokémon, most Legendary and Mythical Pokémon, and a few odd cases such as Nidorina and Nidoqueen. Those Pokémon simply cannot produce eggs, even with Ditto, which is why a perfect Legendary has to be caught or trained as it comes rather than bred.
In play"They share the Field Egg Group, so that pairing will produce an egg."