Waterproof Layer
The waterproof layer is either built into the nappy - like in a Pocket or AIO nappy - or is provided in the form of a cover. There are four commonly used fabrics for the waterproof layer (if we exclude old-style sweaty PVC plastic pants):PUL (Polyurethane Laminate) is fabric (either polyester or cotton) with a thin polyurethane layer bonded to it in a high heat or chemical industrial process. It is more breathable than PVC or other plastic covers. It was originally developed for medical purposes, providing a fabric that is waterproof, comfortable to wear/breathable and durable. PUL fabric is not biodegradable - but you will only require a limited number which you will reuse. PUL with a polyester fabric (poly PUL) is more hardwearing, stretchy and suitable for longer periods than Cotton PUL. 
Cotton PUL tends to come in brighter and more varied prints. 'Minky' nappies have a fluffy polyester outer fabric that either laminated itself, or has a separate layer of poly PUL under it.
TPU (Thermoplastic Polyurethane) is very similar to PUL - it's generally understood by those that use the term that the difference between the two is in the lamination process. PUL is laminated using solvents in a chemical bonding process. TPU fabric has used a high heat bonding lamination process.
Due to this process, TPU fabrics use less chemicals in their production and are more biodegradable than PUL fabrics.
The terminology of PUL v TPU is not standardised, though, and some brands who use a heat bonding process on their laminated fabrics still refer to it as PUL as that is the more recognisable term.
Wool covers are a fantastic option for nights especially. Being a natural fibre it is very breathable and works by absorbing any moisture not absorbed by the nappy itself and evaporating it. Lanolising wool covers will increase their water resistance and will also help neutralise any odours. Wool covers do not need washing every time they are used but can be left to air dry. Lanolising is easy. Find out how here. You don't have to lanolise your cover every time it is washed - just when you notice that the urine smell is not being neutralised anymore.
Absorbent Layer
Cotton is the fabric nappies have traditionally been made of - usually cotton flannelette or terry. Cotton can have problems with providing enough absorbency for heavier wetters, so nappies made out of cotton can become bulky. It is very durable. It is more readily available and therefore cheaper than some other options.
Organic Cotton is the more environmentally friendly choice when using cotton. It still uses a lot of water, but the use of chemicals in production is reduced and it's much safer for the farmers and their families. It is more expensive than ordinary cotton. The generally higher quality, however, often makes it more absorbent as well.
Bamboo can be 60% more absorbent than the same weight in cotton, so nappies of bamboo are trimmer and more absorbent. Bamboo absorbs quickly and traps moisture into the fabric well. It takes the longest to dry. It is the most popular fabric used in cloth nappy making.
Hemp also absorbs much more than cotton leading to trimmer, more absorbent nappies. It also has antimicrobial and antifungal properties. Hemp holds on to moisture very well. It can take as long as bamboo to dry and is not as soft. Hemp can also trap odours in the fabric which can usually be removed by doing a strip wash and by adjusting your washing regimen.
Microfibre is a synthetic fabric - you find it used for cleaning cloths. Microfibre soaks up moisture rapidly (you've seen the ads for the 'magic' microfibre cloths, right?) and can hold a lot of liquid. It is also quick to dry - much quicker than bamboo or hemp. Because the liquid molecules attach themselves to the fibres of the fabric rather than being absorbed into the cells (like with natural fabrics), microfibre can be subject to compression leaks. Because of this, using microfibre with hemp or bamboo is a great combination for long car trips, sleeping or heavy wetters. Microfibre should not be placed directly against baby's skin as it can absorb natural oils and dry out the skin too much.
Stay-Dry Fabrics
Stay-dry fabrics are wicking fabrics that do not absorb moisture themselves but 'wick' it through to an absorbent layer. They are used as the inner layer in many nappies to keep baby's skin dry, or can be used for a reusable liner that is placed in the nappy.
Microfleece is a polyester fabric that wick moisture away to an absorbent layer behind it and stays dry to the touch. Some microfleeces can pill after a period of time, but this shouldn't affect their performance. Microfleece is very popular as a separate liner.