If you’ve hatched one before, that information will be in your Monstipedia. The color narrows it down to a specific species. Its pattern tells you what type of Monster is inside - like these tiger stripes for fantastic Flying Wyverns, or these lame oblongs for Herbivores.
You can tell what’s inside an egg based on its pattern and color. The passive skills that are random are, of course, more rare than the guaranteed genes a Monstie hatches with! Know that some Monsties are born with innate genes that are rare-generally active attack skills. They could be useful when you unlock the Rite of Channeling, which lets you move a gene to another Monstie. Genes with white borders are uncommon, gold are rare, and platinum are ultra rare. Gold shining eggs, and the even rarer Rainbow shining eggs are definitely worth hanging onto. Navirou tells you the smellier and heavier the eggs the better, but there are special effects to look for too.