Kites can be made out of any number of things - usually something light, strong but flexible for the frame, some material that's strong and light for the sail, and some lines (i.e. strings) that are again light and strong but don't stretch. The fundamental goal is to use the lightest material possible for a kite whilst bearing in mind the final price (it's always a trade off I think).
Depending on the type of kites and what's it's going to be used for the actual materials will be different. In many ways there is a perfect material - just the best one for the job in hand at a price everybody can afford.
Various types of wood can be used for certain types of kite - usually single line kites. Bamboo is popular as it's light and strong, but others are possible. The advantage is it's cheap (or should be), easy to work with, light and so on.
Spars of fibreglass at different thickness (anything from 4mm to 8mm diameter are common - others are possible). Fibreglass is relatively heavy and bendy - to make it less bendy the spars will have to be thicker & therefore heavier. On the plus side they are much harder to break and much cheaper. They are generally used in entry level kites or sometimes parts of more advanced kites (such as the C-Quad spars).
The standard material for most kites (certainly all of the ones I have come across). They are strands of carbon fibre running from end to end, wrapped in some kind of resin coating to bind it all together. It's strong, light and doesn't end as much as fibre glass. Again carbon fibre is available in a range of sizes - 4mm to 8mm are the most (especially 6mm) common but as always there are others as well. Usually the spars of hollow to keep down the weight, but solid carbon fibre spars are possible. However carbon fibre spars do break is bent too much and are an excellent way of giving yourself splinters if you are not careful.
Same idea as standard carbon fibre, but instead of the fibre strands going end to end, they are wrapped in a spiral pattern to form the spar. This increases the strength whilst allows the weight to be kept down. However usually these cannot be cut to size and must be used in the length given (not that I have tried but I have heard that cutting reduces the strength of the spar). Often you see 2P, 3P, 5P etc. used when talking about Wrapped Carbon Fibre - these are the number of strands used in the wrapping - the 2P are lighter spars than (say) the 5P & can be used in lighter wind kites. All Revolution kites use this type of spar.
Most (if not all) stunt and traction kite sails are made from a ripstop fabric. Ripstop means a heavier thread is woven in with the others that prevents small holes become large hole - i.e. stops rips. The fabric is also coated with a plastic type substance (on one, or sometimes both, side) that repels moisture and makes it more air tight.
Most of the time we have Ripstop Nylon or Ripstop Polyester - but they go by different trade names such as Ventex or Icarex. Finally these fabrics have different weights - which are given as the weight for a square meter of fabric - g/m2 obviously light fabrics will be more suitable to kites destined to fly in light winds, heavier fabrics for stronger wind kites.
See http://www.kitelife.com/archives/NOVE99/bork-2.htm for some comments on the different types.
Coated on one side only and isn't that water repellent. It is also quite stretchy and will, over time, stretch permanently. It is however the cheapest fabric for kite making and readily available.
Weight: around 42g/m2
Cost: around £4.00/m.
Icarex is a product name from Voltair Technology in the US and was invented in 1992. The fabric is lighter than other fabrics, more resistant to fading (caused by ultraviolet light), good water resistant properties and stretches less. The downside is that it is substantially more expensive than the other fabrics. It feels like plastic tissue paper and it's a good fabric for using on framed kites.
Weight: around 31 to 38g/m2 depending on which type.
Cost: around £11.00/m
Coated on both sides which makes it very water repellent. Less stretchy than Carrington, but more than Icarex. As it's more elastic than Icarex it had accommodate greater strain before it tears (around 5kg as opposed to 2kg). Chikara is used a lot on traction kites (certainly the Peter Lynn C-Quad).
Weight: around 38g/m2
Cost: around £7.00/m
And of course there are other variation upon these themes (such as Ventex - a bit like Icarex but made by the kite maker HQ/Invento) - however these are the types I have come across when looking at kites and looking for kite making materials.
As a final note kites could be made of completely different fabrics - traditional Chinese kites are sometimes made of silk or paper, fighter kites can be made of that metallic wrapping paper stuff - it all depends on what kite you are making and what trade off you are wanting to make.
To be written.