Thirteen in all. The No Egg group at the bottom includes all genderless Pokémon, baby forms of certain Pokémon, and Latias and Latios. As the name says, you can't breed them unless you have a lovely Pokémon called Ditto, but even then you can only breed some of the non-legendary genderless Pokémon for eggs of that Pokémon. Anyway, if you're having trouble finding a Pokémon's egg group, try to compare the physical appearance of that Pokémon to another. Absol is a Pokémon that has four legs and paws to go with them, so a Pokémon in its breeding group should have a similar appearance, like Manectric. Wobbuffet is a Pokémon that relates to no animal or plant, so Koffing can be a mating partner with it.
So after you stick a compatable male and female Pokémon into the daycare, the old man outside outside will tell you the status of your two Pokémon, if they're going to mate or not. Out of the few things that he says to you, there's one thing that he will say that tells you that they will not mate at all. After that, they'll produce an egg depending on, again, what the daycare man says. It could be almost instantaneously to get an egg, or it could take a whole boring day. You will get one though, be patient. This is all dependant on several things, their natures, ID numbers and so forth. It will say that they prefer to play with others if the two Pokémon are not going to breed. A Traded Pokémon for example can take less time to get an egg with than one you personally caught. The rates are as follows;
Same species with different ID numbers:
Chance of getting an egg; 70% | Statement; "The two seem to get along very well!"
Same species with same ID numbers:
Chance of getting an egg; 50% | Statement; "The two seem to get along"
Different species with different ID numbers:
Chance of getting an egg; 50% | Statement; "The two seem to get along"
Different species with same ID numbers:
Chance of getting an egg; 20% | Statement; "The two don't seem to like each other"
Different Egg Groups:
Chance of getting an egg; 0% | Statement; "The two prefer to play with other Pokémon more than with each other."
After every 256 steps, the chance is taken and if the random value is selected to allow for an egg, the egg shall be created for you to pick up at the Day Care centre. If no egg is produced, then you need to go back and partake in another 256 steps.
Hope this helps