Short answers, pulled from the story.
An instrument is classified as a woodwind based on how it produces sound, not its material. Woodwinds either split a stream of air across a sharp edge or use one or two reeds to set an air column vibrating. The saxophone, for example, is made of brass but is a woodwind because it requires a reed.
The two main types are flutes and reed instruments, also called reed pipes. Flutes produce sound by splitting an airstream across a sharp edge, while reed instruments produce sound through the vibration of one or two reeds coupled with an air column inside the instrument.
Single-reed instruments, such as the clarinet and saxophone, use a single reed attached to a mouthpiece with a ligature; air forced between the reed and mouthpiece drives the internal air column. Double-reed instruments, such as the oboe, cor anglais, and bassoon, use two small bound pieces of cane that vibrate against each other as air passes through.
The double-reed technique has been estimated to have originated in the middle to late Neolithic period. Its discovery is attributed to the observation of wind blowing through a split rush.
The modern orchestra's woodwind section typically includes flutes, oboes, clarinets, and bassoons. Supplementary instruments include piccolo, cor anglais, bass clarinet, E-flat clarinet, contrabassoon, and occasionally saxophones.
The concert band's woodwind section is typically much larger and more diverse than the orchestra's. It includes saxophones as standard members and may also feature instruments such as alto and bass flutes, alto clarinet, basset horn, contra-alto or contrabass clarinet, and soprano and bass saxophones.