What happens when Green Lantern's ring runs out of 'juice'?

This answer is from the comics, as they don't show it in the movie. In the comics, classic Green Lantern had to recharge the ring every 24 hours. This (and a weakness to yellow objects) was his kryptonite; that crucial weakness that prevents him from being an invincible god. So, it would be common for fights to go longer than Hal expected, resulting in him running out of charge mid-fight. Then Hal is just a regular guy (but the uniform would stay, even though it's generated by the ring...), and had to use his wits and environment to defeat his foe.

Modern Green Lantern (since roughly the year 2000), instead has the ring be more like an electronic gadget. He charges the ring, and the more he uses it, the faster it drains, until it runs out of power. When he charges it off the lantern, it returns it to maximum charge.

Note that the rings are capable of providing their charge level upon request. So during long battles, you'll see Green Lanterns checking their rings' charge. The modern rings also result in situations where a stranded Lantern will set things up so they're in a low-power-usage state. For example, if they're in space and unable to reach their lantern in time to recharge, they can have the ring put themselves into hibernation, provide a life support bubble, and just drift toward their destination while sending an occasional distress signal. This then helps the Lantern survive until they can be rescued and get back to a lantern to charge their ring.