Common Fur Types

NORMAL FUR
Normal fur is the wild type. It has longer, heavier guard hairs and a shorter, softer undercoat. When stroked backward, it will either snap back ("flyback") or gently roll back ("rollback") into place. Normal fur comes from the absence of any mutations in the fur genes.

REX FUR
Rex fur is characterized by an approximately 43% reduction in guard hair length and 20% reduction in undercoat length compared to normal fur, as well as a decrease in guard hair diameter and altered guard hair shape (Diribarne et al., 2011). Rex also have curly whiskers instead of straight. Simplified, the rex allele is r. This is recessive to normal fur, which is R. In order for a rabbit to have rex fur, it must have two rex alleles (rr). Normal furred rabbits can carry the rex allele (Rr). Rex 
R: Normal fur
r: Rex fur

However, things are more complicated than that oversimplification above! There are actually three known rex alleles, r1, r2, and r3 (Diribarne et al., 2011; Castle & Nachtsheim 1933).. A combination of any of them (absent the R normal fur allele) will result in rex coat. The r1 mutation of the LIPH gene is the most common in rex coated rabbits, so it can generally be assumed that any rex rabbits are r1r1. The r1 and r2 mutations are on the same chromosome and are in linkage, whereas the r3 mutation is on a different chromosome. 
R1: Normal fur
r1: Rex fur
R2: Normal fur
r2: Rex fur
R3: Normal fur
r3: Rex fur

R1R1, R2R2, R3R3, R1r2, R1r3, R2r3 combinations all produce normal fur. r1r1, r2r2, r3r3, r1r2, r1r3, and r2r3 combinations all produce rex fur.

The density of the coat is further influenced by mutations on the following genes: CCNA2, BMP2, IGF-1, TGF-β1, TGF-β2, and MMP2 (Chen et al., 2011a, Chen et al., 2011b, Chen et al., 2017).

WOOL/LONGHAIR
Wool is characterized by both the guard hairs and the undercoat being capable of growing much longer than normal.  This is due primarily to a recessive mutation on the FGF5 gene (Zhao et al., 2018). Wooled rabbits may have wool or normal fur on their ears and face, depending on the breed. When there is wool on the ears and face, it is called "furnishing". The wool allele is l. This is recessive to short hair, which is L. In order for a rabbit to have wool, it must have two wool alleles (ll). Short furred rabbits can carry the wool allele (Ll).

L: Normal fur
l: Wool/Long hair

However, it is more complicated than this. Many other genes are also likely to influence the length of the wool Including WNT5A, WNT11, BMP4, BMP7, MSX2, PDGFA, and IGFBP5 (Zhao et al., 2018). These may all contribute to the final length of wool and account  for differences between individual rabbits' capacity to grow longer or shorter wool.

SATIN FUR
hair shafts which are thinner and slightly translucent, allowing more light through to give a 'sheen' like satin fabric. The sheen typically intensifies the color. Satin fur is characterized by The satin allele is sa. This is recessive to normal fur, which is Sa. In order for a rabbit to have satin fur, it must have two satin alleles (sasa). Normal furred rabbits can carry the satin allele (Sasa).

Sa: Normal fur

sa: Satin fur

MANE
The mane mutation causes longer wool around the head and neck (like a lion's mane) and around the lower part of the body, especially near the haunches (called a "skirt"). The mane allele is is incompletely dominant to the non-mane allele. The mane allele is M. The non-mane allele is m. Rabbits without a mane at any time in their life are mm, and are called no-manes. Rabbits with a single allele for mane and a single allele for no-mane (Mm) are called single mane. Single mane rabbits may shed out most of their mane and skirt upon maturity. Rabbits with two mane allele (MM) are called double mane, and as young kits their sides remain hairless longer than the rest of their body. These hairless areas are where the wool will grow in for the mane and the body wool.

M: Maned

m: Normal (no mane)

COMBINATION FUR TYPES

Rabbits can have combinations of different fur types. Below are different combinations.

Uncommon/Rare Fur Types

FURLESS/HAIRLESSNESS/HAIR REDUCTION MUTATIONS
Rabbits with these mutations are characterized be reduced or completely absent fur. There are multiple mutations which can cause whole or partial furlessness (Geren et al, 2017; Botha et al, 2014; Boucher et al, 1996).

Furless

Furless rabbits are characterized by highly reduced or complete absence of fur. Patches of fur may still be found on the ears, muzzle, tail, and feet but are usually mostly or fully furless over most of the body. The Furless (AKA hairless) allele is f. This is incompletely recessive to normal fur, which is F. Rabbits which are Ff typically bald as kits but eventually grow in full fur. In order for a rabbit to be furless, it must have two furless alleles (ff). Furless allele carriers often have wiry fur, lose their fur easily (in clumps), and/or have bald patches. The Furless mutation often shows up fairly often in mini rex and rex. 
F: normal fur
f: furless

Pelt Loss
Rabbits with the Pelt Loss mutation have hair follicles in the skin, but they fail to emerge from the skin, resulting in thickened skin. There are two genes involved, each with a normal and a mutated version. Variant 1 mutation makes an absence of fur, though less pronounced than the Furless mutation. Variant 2 causes lack of the undercoat hair only, so the coat is thicker than in Version 1.
Ps1: Normal
ps1: Pelt Loss variant1
Ps2: Normal
ps2: Pelt Loss variant 2

References PDFs