Leopard geckos have three types of pigmentation, contained in three different types of cells: melanophores, which contain black pigment (melanin); xanthophores, which contain red, yellow or orange pigment; and iridophores, which contain crystalline materals that reflect and refract light.
Albinism is usually when animals cannot produce melanin because they lack the essential enzyme tyrosinase. Albinism in leopard geckos is apparently slightly different, because they are not true albinos but rather extreme amelanistics, which roughly means that they produce virtually no melanin, but still have the potential capability to do so (because all strains do in fact have tyrosinase). Since it only affects the melanin pigmentation, albino leos can still show orange, yellow and brown colouring.
Tremper, Rainwater and Bell
There are three strains of albinism in leopard geckos, which are each recessive. The "defective" gene occurs at a different allele for each strain. That means that crossing, say, tremper with tremper lines will get you albino offspring, whereas crossing between lines - say, tremper with rainwater - will get you normal offspring.
As for distinguising between the strains, it's not so easy as you might think. Looking at each strain as a whole and comparing it to other strains shows some differences, but since there is a lot of variation in pigmentation and patterning within each strain, looking at a single leopard gecko of unknown lineage may not tell you which strain it comes from. Having said that, there are certain traits you can look for in order to make a good guess - the most distinctive of which seems to be the eyes.
Trempers are the most common type of albino in leos, and also the first discovered. They have pale or silver eyes, often with red veins showing in the eyes. Sometimes the eyes start off pink but turn silver as the gecko matures. Trempers are more likely to have brownish markings on them than the other two strains.
Rainwater albinos (also known as Las Vegas albinos) are smaller than the other two types, and they have darker eyes. They are usually paler in general, with more pink colouring.
Bell albinos are the most recently discovered. Their light pink eyes are their most distinctive feature. They also often have lavender colouring, and small brown spots.
There are other morphs which are not genetically albinos, but might be confused with them.
Patternless (leucistic) geckos have no dark pigmentation but normal-coloured eyes. As babies they have a unique spotted appearance around the head and shoulders.
Blizzards are similar to patternless geckos; typically greyish but could have a background colour of anything between white and yellow.
Hypomelanistic ("Hypos") have a gene which reduces the distribution of the black markings making them have very small number of black markings on the body, but they still have markings on the head and tail. ("Super-hypos" or "baldies" have no markings on the head either.)
Snows are axanthic (lacking the yellow pigment) which means they have a high contrast black on white pattern.
There are also combinations of the above with the various albino strains, creating morphs such as blazing blizzards (albino trait and blizzard trait, making them pure white) or patternless albinos.
Detecting recessive genes
The genes for albinism in all three strains is recessive, meaning that you cannot detect any traits in a "het" or "carrier" gecko. The only way to determine if a normal-looking gecko carries the recessive gene is to breed it with an albino or a known het (making sure that the other albino or het is of the same albino strain as the one you are testing for, of course). If you produce albino offspring, the parent was a het.
Caring for Albinos
Leopard geckos are crepuscular, so are somewhat sensitive to light anyway. However, albinos are particularly sensitive to light and care should be taken, even more so with hatchlings. Turn off room lights when feeding.
Leopard Geckos, by Lance Jepson
Guide to Leopard Gecko Genetics
The Leopard Gecko Manual, assorted experts at Advanced Vivarium Systems
Tremper Albino, Rainwater Albino, Bell Albino at the Leopard Gecko Wiki
Leopard gecko names