(1) "Chasing the dragon" is a method of smoking opium (opium is often referred to as a dragon in asian cultures). The opium is heated on some kind of metal and a fine line of smoke evolves from the heated drug. This smoke is now to be "chased" (meaning breathed in), often through a small tube, like a straw.
(2) Also used to describe a similar method of smoking other drugs (heroine, crack, etc)
In many cultures it is common to chase the dragon every day.