In 1978, NASA astrophysicist Donald J. Kessler theorized that if the cloud of technological debris, satellites, and general human kipple in Low Earth Orbit (LEO)1 grew dense enough, it could lead to chaotic, cascading collisions. In a worst case scenario, this could interfere with human space exploration and satellite communication for decades.
This was, and still is, a purely theoretical problem. It is, however, a real danger looking forward into the future, and is something that is taken into account when designing LEO satellites and manned space flight missions.2
Due to gravity, space junk in general and the Kessler effect in specific are self-limiting problems. If you leave something in orbit long enough, it will fall towards the Earth and burn up in the atmosphere; likewise, if things start bumping into each other, the chances of them falling out of orbit increases3. However, these effects can take a long time to play out, and even with the current low level of LOE space junk (estimated at 300,000 pieces larger than 1 cm), we already lose about 1 satellite a year.
We saw a small example of how this sort of event might play out in 2009, when two satellites collided at high speed -- Iridium 33 and Kosmos-2251 hit each other at a nearly right angle, destroying both satellites and creating 2000 pieces of identified debris4. This was obviously an exciting (and expensive) event, but the fallout was not so terrible; the most news-worthy result seems to have been that the ISS had to make a small collision avoidance maneuver in 2011. Much of the debris are still flying around, although a significant portion have already fallen out of orbit.
While currently most active work in preventing Kessler events involve tracking existing debris and limiting the creation of new debris, there are plans to engage in some serious active debris removal (ADR) sometime in the future. You know, when we have time. These plans tend to look at either deorbiting entire satellites before they break up, or mass removal of small debris between 5 mm and 1 cm. This smaller range is targeted because there is a lot of debris in this range that are just large enough to cause serious damage. Some plans include lasers for small debris, and solar deceleration sails for satellites.
Kessler syndrome is also sometimes referred to as the Kessler effect, collisional cascading, and ablation cascade.
References and notes:
1. Low Earth Orbit covers everything between 160 kilometers (99 mi) and 2,000 kilometers (1,200 mi) in altitude.
2. This includes designing satellites to safely deorbit themselves (or a boost into a graveyard orbit) at the end of their operational life. This also includes jettisoning the contents of space toilets on a trajectory that causes them to fall and burn up in the Earth's atmosphere rather than entering orbit.
3. In addition, below 1,000 km in attitude residual atmosphere starts to produce significant drag, which speeds up orbital decay.
4.NASA: Orbital Debris Quarterly Review, 13:2, April 2009 (.pdf)
NASA: Orbital Debris Quarterly Review, 15:3, July 2011 (.pdf)