Hair is a great ingredient in compost as it adds nitrogen. Large quantities (such as from a hairdresser) could be put in a compost bin covered by a layer of lawn clippings to aid breakdown.
Both human hair and pet hair can be put into compost or a wormfarm.
Human hair can be twisted into string or rope which is wonderfully strong for then knotting or crocheting into bags.
Some kinds of pet hair can be spun into yarn for weaving and knitting, or used for needle-felting.