In the early days of aviation, the pioneers of flight struggled with one major problem - weight. Early engines had difficulty producing enough thrust to overcome the relatively heavy airframes. However, improved materials technology has lead in the last few decades to the creation of a completely new breed of material - the composite.
The actual idea of composites is nothing new. Several ancient cultures would cut strips from different woods, bind them and glue them together with resins to create bows that were light, strong and powerful. However, in modern times, the science of composites has very much taken off. This is because composites offer several advantages over standard metals in certain applications, so their use has expanded from a being a scientific curiosity, useful really only in very high end situations, to a reasonably common part of most high-technology situations, including in the aviation sector.
The most common composite material most people are familiar with is Carbon Fibre Composite. This is manufactured by bonding together sheets of fibres made of the element carbon and some other additives with some type of polymer, usually a type of epoxy resin or similar. Though there are others, carbon fibre is still the most common, and finds applications in areas as diverse as Formula 1 racing cars, the construction of ultra-lightweight canoes and tennis racquets, and indeed in aviation.
There are many applications of composite materials in aviation. Firstly, they are used to add strength or provide support to parts that might be otherwise vulnerable. For example, it is now common for fuel tanks to be armoured with carbon fibre plates that have been made with a blend of carbon fibres and Kevlar fibres, the primary component in most bullet-proof vests. The resulting material is incredibly strong, but also many times lighter than the amount of steel required to provide equivalent protection. The tyres on large commercial jets are now frequently constructed of rubber reinforced with Kevlar fibres; this is to strengthen the tyre and prevent it from breaking with catastrophic consequences, as is believed to have happened in the Concorde crash of July 2000. Composites are also used in other applications where metals would have a negative impact; for example, radar domes are generally built out of composite materials, as the radar waves can reflect off of a metal radar dome inside an aircraft and interfere with the operation of the radar.
Some smaller aircraft and gliders are built almost exclusively out of composite materials such as reinforced fibreglass. This is because they still need to maintain the strength of the airframe, but do not have the incredible power of modern Jet engines at their disposal to provide thrust; in the case of gliders, they have no engine of their own at all. Therefore, composite materials are used here to construct a strong but lightweight airframe; and since these aircraft are reasonably small, they can be built almost entirely out of composites without the cost spiralling out of control.
In general in aviation, composites are chosen due to their unsurpassed strength-to-weight ratio, as well as other benefits that they bring to specific situations. They can be a safer choice than metals in some respects as well; a small crack in a metal can spread with horrific results, but the interwoven fibres in a composite stop cracks spreading before they become a danger.
The main downside of composites is the cost. Though some, such as fibreglass, are now reasonably cheap, the higher-end composite materials used in aviation are still very expensive. Although good for some situations, they do also have their downsides; they are expensive and harder to work with then metals in the construction of aircraft, and they are much harder to repair. Metals can be hammered back into shape and retain all their original characteristics, but a broken composite component must be taken off and replaced if it is damaged.
In brief, composites are an incredible part of materials technology; lightweight and incredibly strong, but still with their downsides.