All creatures begin their lives in eggs. They grow large, hatch and then disappear. In Creatures 1 or Creatures 2, if you or your norns pick them up, they will not hatch unless put in the incubator! In C3 and DS, eggs do not have this problem, although they can be put in suspended animation by being submerged in water or by being put into the inventory.
The gender of a baby norn in C3 and DS has depends partially on the season in which its egg was laid - spring eggs favour girls, and autumn eggs favour boys, while summer and winter give an even split. (This can be disabled with No Seasonal Gender Bias.) C3 grendels like to sniff eggs out and take them home to the Grendel Jungle.
Egg Layers exist in all the Creatures games to make eggs appear at random in your worlds.
In Creatures 1, eggs are occasionally known to appear in the sky.