I'm not sure which parts you don't understand, so I'll just go through all of it!
First there are base colours
, either yellow base or white base. There are two alleles for this gene so a yellow base bird can be carrying a white base allele. If the budgie is a normal yellow base, it will be green, if it's a normal white base, it will be blue. Yellow base is dominant to white base.
: there are two different alleles for them. Dark factors are dominant to normal colour. Normal white base budgies are sky blue and normal yellow base budgies are lime green. If a white base bird is heterozygous (or single factor) for dark factor (only carrying one dark factor allele), it will be cobalt, if it's homozygous (or double factor) for dark factor (carrying two alleles for dark factor), it will be mauve. If a yellow base bird is heterozygous for dark factor it will be dark green, if it is homozygous, it will be olive green.
: the alleles are indistinguishable (heterzygotes and homozygotes look the same). Grey is dominant to normal. A bird with a grey factor is grey green in yellow based budgies and just grey in white based budgies.
: there are two different alleles and it is semi-dominant to dark factor. True violets are white base budgies with either one violet allele and one cobalt (dark factor) allele or two violet alleles and no dark factor. Violet is harder to spot in yellow based budgies.
: there are multiple alleles for this trait, clearwing, greywing, and dilute. It is an example of co-dominance. Normal is dominant to clearwing, greywing and dilute, clearwing and greywing are co-dominant and create the mutation full body colour greywing when combined. Clearwing and greywing are both dominant to dilute. A clearwing budgie has a normal body colour and white or yellow wings with very faint or no markings. Greywings have a 50% diluted body colour and faint grey markings on their white or yellow wings. A full body colour greywing looks like a greywing without the body colour dilution. A dilute budgie has an approximately 80% diluted body colour and very, very faint wings markings.
: has two different alleles, and is dominant to normal. Heterozygotes usually have a strip of clear (white or yellow) feathers across their bellies, and across the bottoms of their wings. They also have a patch of clear feathers on their heads. Homozygotes are almost all clear with a strip of body colour (green or blue) on their upper chest near their neck and sometimes a patch of body colour on or around their rump. They also have all or mostly clear wings and varying amounts of head stripes; they also have a patch of clear feathers on their heads.
: has indistinguishable alleles, is recessive to normal and dominant pied, and is co-dominant with clearflight pied. They have mostly clear feathers except for their lower bellies and their rump. The male's cere will stay a pinkish-purplish colour, while the female's will be normal, their eyes are a dark plumb colour and they don't develop iris rings. This mutation only shows when the budgie is homozygous for it.
: has indistinguishable alleles, is dominant to normal, and co-dominant with recessive pied. They have clear flight feathers and a clear patch on the back of the head.
: a combination of recessive pied and clearflight pied. This variety is all clear, so entirely white or yellow with plum coloured eyes and no iris rings, the male's cere will stay pinkish-purplish.
: sex-linked and recessive to normal striping. With female budgies, this mutation will show when only one allele is present, (since the W chromosome doesn't hold and colour genes) but with male budgies, opaline will only show when they have both alleles. Opaline budgies have thinner black stripes on their head and the back of their neck, because of this, their body colour can be found between the stripes on the back of the neck and into the wing feathers a bit. Their tails also have some body colour running down them and their flight feathers have clear areas.
: two different alleles, and dominant to normal. A heterozygous budgie will have faint head stripes, clear wings with a light black outline around each feather and clear tails. A homozygous budgie will be completely clear, but they will have iris rings and normal coloured ceres.
Yellowface type I
: also known as creamface, and it's dominant to normal... Yellowface type I is a combination of two different genes that both code for a broken yellow-producing enzyme which doesn't create enough yellow to make the bird green; these two genes compensate for the other's faulty yellow production and together they create just enough yellow to make the face a pale cream colour. A homozygous creamface budgie just looks blue because it either has two alleles coding for one or
the other of the broken enzymes, not both. A creamface budgie is blue with light yellow face and some yellow is the tail feathers.
Yellowface type II
: two different alleles, and it's dominant to normal. A heterozygous budgie would have a bright yellow face, yellow in the tail feathers and a sea green/turquoise body due to the yellow spreading all the way down the body and mixing with the blue body colour to create turquoise. A homozygous budgie would have a bright yellow face, yellow in the tail feathers and sea green/turquoise body colour that only moves to the upper chest. Usually the turquoise colour appears after the first molt. It's actually a yellow reducing mutation, which is why the heterozygotes have more yellow in the body feathers than the homozygotes do.
: different alleles, and is dominant to normal. There are two types of single factor in goldenface; the first type is slightly brighter than creamface and is restricted to the face and tail feathers, the second type is just like single factor yellowface type II, but with brighter yellow and darker sea green. A homozygous goldenface budgie has a vibrant golden face, and sometimes the yellow moves to the upper chest too.
(albino or lutino): sex-linked, recessive to normal, and co-dominant with cinnamon. Albinos are all white with red eyes while lutinos are all yellow with red eyes. Female inos only have one allele for it, while male inos must have two alleles for it in order to show it. Ino masks other mutations; the only mutations not masked by ino are cinnamon (creating the lacewing variety) and yellow/goldenface (creating the creamino variety). The male's cere will stay a pinkish-purplish.
: sex-linked, recessive to normal, and co-dominant with ino. This mutation causes the head stripes and wings to turn a light brown colour, it can affect the body colour too. Males need two alleles, while female need only one to show the mutation.
: sex-linked, recessive to normal, a combination of ino and cinnamon. These budgies are white or yellow with light brown cheek markings, head stripes, wing markings, and tail feathers. Their eyes are red and the male's cere will stay a pinkish-purplish.
: sex-linked, recessive to normal, but dominant to ino. These budgies have completely clear or diffused body colours that get darker towards the lower belly and rump. They have light head stripes and dark wing markings that slowly fade to a lighter grey colour towards the ends of the wings.
: recessive to normal. The body colour is mostly clear and becomes less diluted towards the lower belly and almost normal body colour on the rump. Their head and wing markings are a light brown colour. They have red eyes and the male's cere stays pinkish-purplish.
: is not
an inherited mutation! I thought I'd add it in here anyways. A half-sider is a tetragametic chimera, resulting from the fusion of two fertilized zygotes. They are most striking when they are the combination of a white base budgie and a yellow base budgie split vertically right down the middle. They are often hermaphrodites because one zygote was female while the other zygote was male. These budgies often go undetected when they are the combination of two zygotes of the same colour.
Okay, phewf! Those are all the mutations I could think of off the top of my head.
I hope it helps you out!