What materials are planes typically made of and why?

One of the most important objectives for airplane designers is reduction of weight - build something that meets all the functional requirements while weighing as little as possible (within reasonable cost parameters, of course. Unicorn skin is way too expensive, for example. There's a joke that a commercial airplane designer would sell his grandmother to save 100 pounds of weight from the plane. ). Less weight means less engine thrust is needed, which means less fuel is burned. Less fuel burned means the airline makes more money. Airlines love to make money :-)

Large commercial transport aircraft have traditionally been made with aluminum skin over an aluminum frame. Aluminum is a great metal for airplanes since it's light and can be pretty strong when alloyed. But it has some issues over time, including corrosion and metal fatigue. With the introduction of the Boeing 787 a few years ago, large-scale use of carbon fiber reinforced plastic, aka composite, is proving itself as a lighter weight, corrosion-proof, and stronger replacement for aluminum skin. The Airbus A350 entering service later this year also is skinned with composite material, but uses panels over a frame rather than one piece composite fuselage barrels like the 787. Here's a video of a 787 fuselage section being built up:

As you can see, this is a very different way of building big airplanes.  The chamber seen later in the video into which the fuselage barrel is inserted is an autoclave, basically a large oven that is used to cure and harden the resin used in the composite material. When the barrel is complete, windows and other openings are cut into it as needed. 
Most of the stuff on the inside of the plane is made from aluminum or plastic, with artificial fibers used for carpets and upholstery.

A350 is another engineering marvel with 52% Composite Material , which includes Glass Fibre / Carbon Fibre.  With the reduction in weight they have increased the fuel efficiency by 4% which is a huge achievement .
Further to these every fastener that goes into the aircraft is either made of Titanium / Aluminum / Steel,However the composition of each is specific to the manufacturer , namely Airbus , Boeing , etc.
On the other hand smaller planes can also be made of light weight wood - used only for bush flying.