First things first: regular, unmutated peachfaced lovebirds come in two base flavors: green series (which is dominant) and blue series (which is recessive). Peachfaces have the most mutation possibilities out of any other lovebird breed, and these mutations fall under recessives, dominants, partial-dominants, and sex-linked. Blue-series has two variants: Dutch Blue and Whitefaced Blue.
In lovebirds (as well as other birds and some reptiles), instead of the chromosome setup being the male having XY and the female having XX, it's the other way around. Since the female is the one carrying the sex-determining chromosome pair, the offspring will usually only have one sex-linked mutation. There is a very slim chance of two sex-linked mutations occurring, and if that does happen, then both mutations will be on the male's chromosome pair.
Female lovebirds are naturally "visual" when it comes to sex-linked mutations, meaning that if they're carrying the mutation, they'll show it. If she looks as though she doesn't have the mutation, then she doesn't have it. A male, however, can be "split", still carrying the gene for the mutation but not showing it.
While color intensity and specific patterns vary bird to bird, these are the basic traits found within each of these mutations.
Okay, got all that? Good. Let's begin.
A Dutch Blue is a dark-eyed, mostly-blue bird with a white-gray face and an orange strip across its forehead. It is considered to be the quintessential blue.
Whitefaced Blues are often confused with and sold as Dutch Blues. Like the Dutch blue, they are mostly blue, but they mostly lack even the faint red hues the Dutch blues. They still have all the genes for the reds and yellows, and usually average birds have the faintest traces orange above their eyes, but the ideal show bird wouldn't show the color.
A Seagreen mutation is the result of crossing a Whitefaced blue and a Dutch Blue. The bird's head is pretty much the same color as a Dutch Blue (white face, orange visor, dark eyes), but its body is closer in color to that of a regular green bird.
A Lutino mutation (AKA xanthochromism) is a fairly common is a sex-linked recessive mutation in peachfaces that removes all the melanin from the bird's feathers and leaves just the carotenoids (which produce reds and yellows). The end result is a red-eyed yellow bird with white or yellow tailfeathers and a reddish face. A lutino mutation happens when the bird's base color is green.
Creaminos are what happens if the base color for what would have been a lutino mutie is blue-series rather than green-series. Creaminos are very pale yellow (the closest you can get to an albino peachface), have red eyes like the lutino, and usually have a very pale band of red/orange/pink on their foreheads.
Cinnamons are more sex-linked mutations and come in two flavors. A green-series Australian Cinnamon (AKA Pallid) is like the Lutino, but a little darker in coloring (although lighter than the American Cinnamon). If it is an Australian blue-series, it's going to be pale, but the colors that are there will be more vibrant than those of a creamino. Australian Cinnamons have dark eyes. American Cinnamons are usually a medium-green, lighter than the usual dominant green unless they're blue-series, in which case they're pale.
The Orange Face mutation is a regular old recessive mutation that delivers what is exactly on the tin: peachfaces who are orange-faced. Again, color intensity varies, so sometimes the faces are vibrantly orange! sometimes they are very pale orange, and other times they're almost red. Since this mutation is not a sex-linked one, it can pop up with a bunch of other mutations, such as orange faced lutinos or cinnamons.
An American Yellow (AKA Golden Cherry, Dilute, and Yellow) is very much like a Lutino in that it is a mostly-yellow bird with a red face. Unlike the lutino, it also has a pale blue tail, and the yellow is slightly duskier than that of the lutino.
The Opaline mutation is a fairly recent one (1997) and while other mutations change the coloring of the bird but leave the areas of coloration relatively untouched, the opaline mutation actually changes where the colors present themselves. Rather than a peachface having a peach face, it now has a peach head (with slight color alteration behind the ears). The edges of the feathers are lighter than usual, giving the bird a marbled look, and the tail feathers have much more red in them. Opaline is a sex-linked mutation. They can also be combined with other mutations, such as lutino.
A Fallow is often mistaken for an Australian Cinnamon, but its eyes are red like that of the lutino and its blue tail is different than that of the Cinnamon.
Dark Factor is a semi-dominant mutation, meaning that if a single parent has the gene, the mutation will appear in the baby; that is called a single dark factor. If both parents have it, then the baby will be a double dark factor and be twice as dark. Jades are single dark factor greens, olives are double dark factor greens, cobalts are single dark factor blues and slates are double dark factor blues.
Violets are birds that shift from blue to purple. Depending on dark factors, they can range from a light, mostly-blue lavender color to a rich purple.
This is just scraping the tip of the iceberg here, people. Mutations can pile on top of each other and when those mutation combinations they receive their own special name. There are a few places online to calculate and sort out your chances of winding up with whatever mutation you like.