Yes, but here's what that means.
The "sensor" is actually protected by a couple of filters. Typically there's a UV/IR filter and a low-pass filter (depending on camera model) and those two filters are directly in front of the camera sensor. "dust" collects on the outermost filter (it isn't actually possible to touch the sensor itself).
The "sensor cleaning" system is actually connected to the front-most filter and it uses a piezoelectric effect to force that sensor to vibrate rapidly ... hopefully shaking any dust loose (point the camera toward the floor so the dust falls away).
It's not foolproof. Sometimes a static charge will build up and the dust *really* wants to cling to that filter. It is possible to put the camera into a manual dust-cleaning mode (this really just flips up the mirror and opens the shutter so that you can use cleaning tools). To clean manually, you'd normally have a hand-squeezed air-bulb (such as a "Giottos 'Rocket Blower'") and should avoid using cans of "compressed air" (the propellant in those cans leaves a hazy residue on the filter and then it has to be cleaned with more aggressive methods). The "wet" cleaning method typically involves using a cleaning swab and a few drops of a cleaning solution. "Eclipse" cleaning solution is popular and probably in stock at any camera store. It's actually a solution of nearly pure methanol -- because methanol evaporates VERY quickly and leaves virtually no residue behind.
When you swap lenses, take care to protect the camera from dust. When you perform an automatic self-cleaning cycle you can point the camera at the ground (so the dust falls "away").
The "automatic" cleaning cycle may not necessarily run _every_ time you power the camera on or off. Some cameras count how many shutter releases have occurred and wont bother to perform the cleaning cycle if it isn't needed.
