What emergency situation can pilots encounter in gliders/non-powered aircraft?
The number one emergency in a glider is a tow rope break at low altitude, less than 200 feet above the ground with no appreciable headwind. Below 200 feet, the proper technique is to land straight ahead, which usually means landing off airport. 

Between 200ft and 500ft, it is possible to make a U-turn back to the runway, and land in the opposite direction in which you took off.  The main hazard with this kind of landing is landing with a tailwind, and once the wheel touches the ground, the flight controls operate in the reverse direction.

Above 500 feet, it is possible to fly an abbreviated pattern and land on the runway you took off from.

Add 50 feet of altitude for each 10 knots of headwind.

Some other tow emergencies are inability to release to tow rope, in which case the tow pilot needs to release the rope on his end; engine failure of the tow-plane; and a double tow rope release failure, which means the glider needs physically break the tow rope (it has a pre-determined breaking strength of 80-200% of the weight of the glider) or to land behind the tow-plane.

You can also run out of altitude and have to land in a field, which is called an "off airport landing." Someone has to come and get you with a trailer, and you disassemble the glider and drive it away. One hazard is the off airport landing in a field, and the farmer comes out with his shotgun.

Other hazards are flying into the clouds or into severe turbulence, which can happen when the glider pilot is flying "wave," a kind of lift generated by strong winds in proximity to
 mountains.

When thermaling, you are flying very tight steep turns, often with other sailplanes who are also trying to gain altitude. Everyone is supposed to fly the thermal in the same direction, and there is the hazard of colliding with other gliders.

Reference: FAA Glider Flying Handbook - Abnormal and Emergency Procedures