I won't claim huge expertise, but I've done a few courses on the subject. At least I can start things off, then the real experts can elaborate on the interesting bits.
A composite material is defined as one made up of more than one basic material, they're used in various places - even wood, technically, is a composite (of cellulose and lignin). The most common aerospace composite is FRP or fibre-reinforced-plastic, which covers a fairly wide range of materials. there are others, such as MMC - metal matrix composites, which are starting to be found in some of the more complex parts of engines. Re-inforced concrete is a composite material too, so they're nothing new.
An FRP consists of two parts, a fibrous element, which might be glass, carbon, kevlar (which is really a high strength nylon) or some other more exotic material. Older FRP structures used a random array of fibres which were overly heavy and bulky but had the advantage of giving the same strength in every direction. Mostly now the fibres start as a woven mat, which is much more efficient, but doesn't give the same strength in every direction, which means that the designer has to understand both his material, and the loads it'll be exposed to, very well indeed. This is one of the reasons why the material is so expensive.
The other part is the matrix, which is a plastic surrounding the fibres, and can be either a thermoplastic (one which will melt in heat) or far more likely a thermoset (one which is originally set by heat and ultimately will burn rather than melt). This holds the fibres in position, and the real strength of the composite comes from the junction between the two parts, rather than either individual material.
Composite components can be made in virtually any shape and they can be very light because the designer only needs to provide strength where it's really needed. A metal component by comparison provides strength in every direction, rather than just where it's loaded - in other word there's excess weight there. But, if somebody else in the design team changes the load paths, you are potentially in trouble with a composite component; that's how the EH101 at Westlands lost it's tail rotor, causing the crew to learn that it is possible to parachute from a helicopter !
Failure modes are very complex. Unlike metal fatigue, which is about single crack growth, composites degrade through tiny cracks within the matrix, which reduce the stiffness and ultimately the failure loads. Also any impact starts cracking, leading to what's known as "BVID" or Barely Visible Impact Damage. BVID will cause a step reduction in material strength in the impact area. Frankly nobody understands these well, for which reason all the airworthiness codes require much higher strength from composite components than they do from metal ones - usually 25% to 50% more.
When a composite component does fail, it aint like a metal. It keeps appearing perfectly intact right up to the point of failure, then goes almost instantly - it's not like metal parts which start developing bends. If you remember high school physics terminology, the Hookes law period extends right up to the failure point, there's no plastic deformation.
Finally, there's one area often forgotten about. What happens if a composite burns? The answer is release of huge numbers of very damaging particulates into the surrounding atmosphere, I sat through an RAF fire service briefing on Eurofighter once, if one of those crashes and burns the firemen are going in in NBC suits to protect themselves from the tiny particles (not the toxic gasses, which are there but rather less serious).
Hope this helps a bit, if you're interested in materials in general there's an excellent book called "The new science of strong materials" by JE Gordon that I'd recommend. It's written for the non-engineer, and therefore blessedly free of both maths and molecular structure diagrams, which materials scientists usually seem overly fond of. Gordon does talk a fair bit about aircraft structures, although I'm not sure if he covers the most modern aircraft materials; my copy is about 20 years old.
G