Basic Breeding Percentages
Okay basically there are four categories of Ponies - Common, Uncommon, Rare, and Super-rare. In each category are various species. The Species post has all pony species separated into these categories, and they are also listed below.
For breeding purposes, the percentages below are based on CATEGORY and not individual species. Whatever category is rolled for each baby will determine the baby's species - according to parent species. If the percentages do not add up to 100, whatever leftover percentage there is will be Common, and babies rolled with that percentage will be Earth babies.
In any breeding, ponies that have edits but are not listed as Cosplays can still pass down edits - but this is by colorist choice only. Edits on ponies that aren't Cosplays are not rolled for.
Throwbacks are discussed below, and there are also several example breedings to look over.
For Legend Breedings, please see this thread, as those are done slightly differently.
Okay basically there are four categories of Ponies - Common, Uncommon, Rare, and Super-rare. In each category are various species. The Species post has all pony species separated into these categories, and they are also listed below.
For breeding purposes, the percentages below are based on CATEGORY and not individual species. Whatever category is rolled for each baby will determine the baby's species - according to parent species. If the percentages do not add up to 100, whatever leftover percentage there is will be Common, and babies rolled with that percentage will be Earth babies.
In any breeding, ponies that have edits but are not listed as Cosplays can still pass down edits - but this is by colorist choice only. Edits on ponies that aren't Cosplays are not rolled for.
Throwbacks are discussed below, and there are also several example breedings to look over.
For Legend Breedings, please see this thread, as those are done slightly differently.
Categories:
Common: Earth, Unicorn, Pegasus, Flutter, Breezy, Pixie, Demonex, Sea Pony
Uncommon: Cosplay, Demon, Vampire, Mummy, Hybrids, Hippogryph, Other (any other species of pony that is not official, such as Fairy, etc)
Rare: Dragon, Mer-Pony, Kirin
Super-rare: Seryph
Legend: Category by themselves XD. For any breeding with a Legend as the direct parent, please see the Legend Breeding thread.
% of chance that a pony will pass its genes on (PER PARENT):
40% Common
30% Uncommon
20% Rare
10% Super-Rare
For gender rolls, the standard is 50% chance male, 50% chance female. Exceptions are listed below.

Special Breeding Rules
-- Sea Pony x Earth Pony:
80% Common (Earth or Sea Pony)
20% Rare (Mer-Pony)
-- Seryphs:
Seryph x Seryph will ALWAYS produce twins. As in, only two babies. For these twins, there is a 75% chance of babies being male. In the case of RP'd breedings, if there is enough RP for one extra baby, there MUST be enough RP for two - to keep the twins rule. So if you're trying to RP for extra Seryph babies, you will need 10 RP's total to get ANY extras at all.
In the case of a Seryph breeding with any OTHER species, if a super-rare baby is rolled (a Seryph), one other baby in the breeding automatically also becomes a Seryph (again to keep with the twins rule). If it so happens that enough super-rares are rolled and there are not enough babies in the breeding to have a twin, an extra baby must be added to keep to this rule.
-- Kirins:
If parents are BOTH Exalted Kirins, any Kirin babies will also be Exalted - However, they are only Exalted in adult stages and must be sent to Seph to have the wings added. There is one more thing concerning Exalteds, but it is for Staff only to know. Please see this post for more, thanks! Exalted babies are allowed to be customed-in ONLY if there is an Exalted Kirin in the family line.
-- Hybrids:
To custom in Hybrid babies (of any species combination), there MUST be a Hybrid pony in the family line. If Uncommon is rolled in a breeding with a Hybrid, it will be colorists' choice if they want to make a Hybrid pony and what species to combine from the parents/ancestors.

Ancestors & Throwbacks
In the case of ancestors and throwbacks, you will still be using the basic rules above, but only for uncommon-super-rare. Common ancestors will not matter, except for possibly being different Common species. If there is the chance for more than one species in ANY category, it will be colorist choice which species the baby is, within that category.
For throwbacks, each ancestor that is in a certain category will add more of a percentage (per pony) for that particular category, as follows:
15% Uncommon
10% Rare
5% Super-Rare
2% Legend
Commons are done just a little bit differently than the above, because after some time it would be logical to be able to selectively "breed out" Common babies. Also, if there were a ton of Common ancestors, Common babies could end up being bred far more often than would really make sense. As such... there are benchmarks for the Common percentage.
1-5 Common ancestors - 20% each
6-10 Common ancestors - 10% each
11 and up Common ancestors - 5% each
To roll a breeding with throwbacks, you would simply continue adding everything up until you got to a final number. If that number was LOWER than 100, you would add any leftover percentage to Common. If it was HIGHER than 100, you would take that number and use the Random Number feature to determine each baby's category - then choose species within the category rolled out of the family pool.

Example Breedings
Earth x Pegasus (Common x Common)
All Common babies, colorist choice which species each baby is - Earth or Pegasus.
Unicorn x Cosplay (Common x Uncommon)
40% Common, 30% Uncommon,
1-30 = Uncommon baby (edits passed down from Cosplay)
31-70 = Common (Unicorn)
71-100 = Common (Earth)
Flutter x Dragon (Common x Rare)
40% Common, 20% Rare, 60% leftover
1-20 = Rare (Dragon)
21-60 = Common (Flutter)
61-100 = Common (Earth)
Cosplay x Mer-Pony (Uncommon x Rare)
30% Uncommon, 20% Rare, 50% leftover
1-20 = Rare (Mer-Pony)
21-50 = Uncommon (edits passed down from Cosplay)
51-100 = Common (Earth)
Now let's see some examples with Ancestors....
Earth x Flutter, with one Dragon ancestor (Common x Common with one Rare ancestor):
Normally all babies from this breeding would be Common, with colorist choice as to what species. However, there is one Rare ancestor! So the following would happen...
1-90 = Common (Earth or Flutter)
91-100 = Rare (Dragon)
Vampire x Kirin, with two Cosplay ancestors (Uncommon x Rare, with two Uncommon ancestors):
30% Uncommon, 20% Rare + 15% per Uncommon ancestor (15 x 2 = 30)
SO
Uncommon percentage now becomes 60%
1-60 = Uncommon (either Vampire or Cosplay edit traits passed down)
61-80 = Rare (Kirin)
81-100 = Common (Earth)
Okay, let's try something really complicated....
Seryph x Demon, with one Vampire ancestor, one Mer-Pony ancestor, and one Legend ancestor (Super-rare x Uncommon with one each Uncommon, Rare and Legend ancestor):
10% Super-rare, 30% Uncommon, + 15% Uncommon, 10% Rare, 2% Legend
SO
10% Super-rare, 45% Uncommon, 10% Rare, 2% Legend, 33% leftover
1-45 = Uncommon (Demon or Vampire)
46-55 = Rare (Mer-Pony)
56-65 = Super-rare (Seryph)
66-67 = Legend (Earth which grows to Legend after adult stage)
68-100 = Common (Earth)
Also, if a baby was chosen to be Seryph with this breeding, one other baby would automatically be another Seryph, and its twin. The gender rolls for those babies would be 75% chance male, 25% chance female.