Airplanes are amazing. Case in point: last Friday a plane landed safely after one of its engines quit on the way from Boston to Houston (link courtesy of Erik). The plane was a Boeing 737, which only has two engines!
Catastrophes get a lot of the attention, but they're relatively infrequent. Incidents like Friday's are far more common. Something could have gone terribly wrong, but the system was designed to work even when problems come up.
The engineering that goes into airplanes is amazing. Critical systems are either redundant or are designed with large safety factors. Engineers analyze every failure mode and design systems to account for problems that may arise.
About a year ago, I talked to a pilot who flies for United, and I learned something cool: FAA regulations prohibit pilots from landing planes manually in certain conditions, such as in fog or high cross winds. The pilots are actually required to let the autopilot land the plane! So next time you're landing in fog, just remember that it's not that handsome pilot up front, but rather some pudgy Boeing engineer in Seattle who's getting you home safely. :) (Of course, it's still good to have a pilot on board; no computer could have done something like January's Hudson River landing.)
To illustrate what planes are designed to withstand, here's a video of a test that Boeing did for the 777. They calculated the maximum load that the wings could experience during flight, and then designed them to withstand 150% of that load. Next time you see the wings bounce when your flight hits a little turbulence, just remember that they're designed to withstand this: