First, How to Breed:

You can easily see if a pet is available for breeding if you click on the pet. It will show a "breed" button if it's available for you.
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This button will show both on your own pets, and on friends pets that are available. Keep in mind, friends may require something in return, or simply have the breeding status set so they can easily see what pets are "ready". (more on this in a moment)

Female pets have a "cooldown" between breedings- a time where they must rest and recover before they can breed again. While they are on cooldown, the breed button will not be present. Cooldown times vary between species, please see the species page for details. If a pet has been set to "breedable" status (meaning friends can see the breed button and request breed trades), then the pet will also display a small heart when viewed in enclosure mode. This heart will be red for ready, and blue for cooldown.
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Some pet owners set their pets as "breedable", although they have no desire to receive breeding requests from others. They do it simply to see this heart icon for easy cooldown reference. To turn on or off the breedable status, click edit and click the breed button then save your change.

Once you have found the pet you wish to breed, click on the 'Breeding' button shown in the top image. This will pull up a box showing all available opposite gender that reside in the same enclosure as the pet you are breeding. (If you are requesting a breeding trade from another person, it will instead pull up a trade window, with possible pairings below. Drag a pair up into the trade box to proceed.)
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Note the "Angel Project" drop down box. This defaults to the enclosure the pet you are breeding is in. If you have no available pets to breed with that pet in the enclosure, it will be blank! You can click the arrow to select another enclosure to find the match you want.

Once you select the match, you will get a confirmation pop up window. Clicking ok will create an egg between the 2 pets. You will automatically be taken to the egg's detail page. There, you can turn the egg (more on that later), view it's pedigree, splice it in the lab, set it for trade, edit it's presentation, even discard it! A discarded egg is not recoverable.

Hatching Eggs:

When you first create an egg through breeding, you will be taken to it's details page. This is the page you need to be on to turn eggs. However, you can only get credits from turning friends eggs, so many people will not turn their own eggs until they are at 100% and ready to hatch.
An egg that is ready to turn can be seen in hatchery mode by 2 green arrows:
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Please note, in the image above, only the owner of the egg will see the arrows icon. This is because the eggs are ready for the final hatching turn, which only the owner can do! For all other turns, anyone who can view the hatchery can see the icon.
An egg needs to be turned 5 times in it's life. The first happens at 0%, that initial turn to start it on it's way to hatching. After the first turn, the hatching indicator will slowly climb (requires refreshing to see the change, as well as time passing). At 25%, 50%, 75%, and 100%, it will stop, waiting to be turned again before the progress meter will continue. When you see an egg at 24%, you can be ready to turn it soon! Each time the egg is turned, more cracks will appear in it's shell. This is a quick way to see how far the progress is from the hatchery view.
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Once the egg reaches 100%, the button changes from "turn egg" to "hatch egg". At this point, only the owner may turn it. Please keep in mind, once you hit "hatch" you cannot undo! This means, any splicing you wish to do must be done before hatching!
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Egg Ninjas:

Egg ninja is a term for people who come to your hatchery and turn eggs for you. This is benificial to them, as they may find credits while doing so. It neither helps nor hinders you, unless you are a person who doesn't turn your own eggs out of kindness to friends, then it helps you get the eggs progressing again! You can post in your public feed asking for ninjas, or in groups such as this one: http://ovipets.com/#!/Egg%20Ninjas

Genetics and Dominance:

There are genetic research trees, cost, and dominance charts on each species page. This area will discuss how genetics work. Please do read the FAQ on Ovipets, as they have already done a fine job explaining it. This section is for people who may want a little bit more detail. Please note, all of the following information applies to offspring who are unspliced. Details on splicing are below.
First off,
http://en.wikipedia.org/wiki/Punnett_square

Imagine that there is a gene for every slot in the game called 'nothing' and all starter pets have 'nothing' in each square. That should help people understand a bit better. Also, "nothing" is the most dominant gene in the game. If a pet carries 'nothing' and 'gene1', 'nothing' will always be the visible trait!

A pet will carry a gene if it's parent show it. They may also a gene because it's grandparent or great grandparent shows it (or even father back!). Unless you can prove it through unspliced offspring, you don't KNOW a gene is carried unless the parent actually shows it.

If a parent shows a gene, there is 100% chance offspring carries a gene in that slot. Not necessarily the one shown, for example if grandma had batwings, grandpa had freatherwings, and mommy shows featherwings, the baby is 100% to carry some sort of wing, but it could be carrying the recessive batwing that mom carries under her shown featherwings!

Now, to continue down the line, the grandchild of a shower (child of carrier) has a 50% chance to carry. The great grandchild has a 25% chance, the great great grandchild has a 12.5% chance, and so on. The moment you prove a carry through breeding, you can pop that back up to 50% chance for that proven carrier's offspring and start again from there. (as well as confirming that the parents carry back up the ancestral tree all the way to the showing ancestor) I have not received confirmation of this, but I suspect that the chance for genes to pass on is actually quite a bit higher in Ovipets than real life, which this information is based on.

A pet that shows genes will pass those on to the offspring. However, if the other parent has a 'nothing' gene showing, it can pass on as well! This is why a max-vis pet bred with a starter will have no genes showing!

A pet will *never* have more then 2 genes in a single spot. For example, if mother shows fox tail, and father shows nothing, one grandparent shows husky fox tail, another grandparent shows triple solid tail, and so on- ONLY 2 genes make it to the offspring. One from the father, and one from the mother. So, if the offspring of the above pair shows no tail, you know for certain it inherited 'nothing' gene from the father, and you can ignore all other tail genes on the father's side. Only breeding can prove which gene was passed on from the mother's side, unless she was spliced (see below) in which case she passed on what she shows for certain.

Anyone advertising that a pet carries multiple genes in the same location is misinformed. Odds are they are not attempting to scam, genetics are a complicated subject that many misunderstand. Thankfully, some groups such as http://ovipets.com/#!/Auction%20House have set the format standard on auctions and sales to include proper terminology of 'might carry' instead of 'does carry' in such cases.

Dominance:

Dominance is just used to determine what is recessive or dominant. Unlike some similar genetic games ( ponyisland.net for example), it's not a 100% thing that one gene will show over the other, it's more like real life, where a recessive gene may poke it's head out.

For example, feather wings on one side, dragon wings on the other (for a lupus) dragon has a lower dominance, so most times the feather wings will show. However, there's always a chance that will reverse. The higher number, the more likely it is to 'dominate' the other gene. The closer the 2 genes dominance are, the more random chance has a hand in which one dominates.

Splicing:

Before an egg is hatched, it can be taken to the "lab" with the lab button on its detailed view. The lab allows you to splice a single gene that you have fully researched into the egg for a credit cost (costs vary based on gene and level of research for that species). A splice guarantees the mutation will be visible on the offspring. It also negates any carried gene from farther up its family tree in that slot. For example, splicing triple fox tail onto a lupus will give an offspring that both shows and carries triple fox tail. If the mother had husky solid tail and the father had double fox tail, those genes no longer matter. The have been overridden and replaced by the spliced gene. Spliced genes only effect the loci, or area, of their own gene. For example, that triple fox tail does nothing to wings or body.