P̶o̶k̶e̶m̶o̶n̶ ̶E̶g̶g̶s̶ ̶a̶r̶e̶ ̶t̶h̶e̶ ̶b̶e̶s̶t̶ ̶w̶a̶y̶ ̶t̶o̶ ̶s̶t̶a̶r̶t̶ ̶t̶h̶e̶ ̶d̶a̶y̶:̶ ̶W̶h̶o̶l̶e̶s̶o̶m̶e̶ ̶a̶n̶d̶ ̶n̶u̶t̶r̶i̶t̶i̶o̶u̶s̶.̶
Just kidding. Pokemon eggs are mysterious things that appear when certain Pokemon are kept in the daycare together. There is a bird and bees joke somewhere here, but the games strictly maintain a We don't know how it got here kind of attitude and I'll stick with that for now.
Pic from AWKWARDZOMBIE.
That said, there is a some logic behind Pokemon eggs. There are a way to obtain Pokemon apart from catching them in the wild, and allow access to many baby Pokemon (like Pichu or Igglybuff to name a couple), that are unavailable in the wild.
Pokemon eggs are specifically important because they allow you to get a Pokemon with the kind of Nature and IV's of your choice.
This is important for hardcore battlers who look to perfect the art of Pokemon combat.
Back to what eggs basically are, there are a 'prevolution' of sorts to Pokemon which can be hatched by walking a certain number of steps (Different for different species of Pokemon), more on which is here.
Now, the kind of Pokemon you get from the egg is simple: It is the female Pokemon that you left in the daycare.
For an egg to appear, you must deposit two Pokemon of opposing genders but of the same egg group.
If two Pokemon are of different egg groups, there is no way you'll get an egg.
After you receive an egg (which is randomly generated for every 100-500 steps you walk/cycle), you must walk/cycle it around to hatch it, as mentioned above. Of course, the steps can be halved by having a Pokemon with the ability Flame Body or Magma Armour on the team.
(Quick reminder here that the Database's guide on it pretty covers everything on grounds of basic principles, and that you must check it out)
BUT WAIT THERE'S MORE
There are also things known as easter eggs in the games, which are fun little secrets hidden by developers to be found by players. Funnily enough, none of them have anything to do with chocolate or sundays, so that's kind of a bummer.
Okay, that was a joke.
Honestly though, there is a LOT to breeding and eggs, hiding behind the shell and it cannot be covered in one answer. Besides, there are multiple egg-cellent guides online that cover in much more detail, which I suggest you take a crack at (ha ha). And then, if there are still any cough specific cough doubts, feel free to ask them on the Pokebase.
Half hearted apologies for the puns, and a sincere hope this answer helped.