Is the flute woodwind or brass?

Being a woodwind instrument does not necessarily mean the instrument has to be made of wood (although the Indian flute is).

Yamaha says 'The term "woodwind instrument" refers to wind instruments whose sound production principle is not lip reed; lip reed is sound production system of brass instruments which make sound by vibrating your lips. Flute and piccolo produce a sound by vibrating air when you blows air against an edge of embouchure hole on the headjoint. Saxophone and clarinet produce a sound by vibrating a single reed attached to the mouthpiece. Oboe and bassoon produce a sound by vibrating two reeds, so they are called double reed instrument.'

Here is a list of the woodwind instruments.