A complementary filter is the combination of two filters that, in some way, sum together to form a whole.  One popular use case is to use a low pass filter on one input signal and a high pass filter on a different input signal.  When these two are summed together, the resulting signal has both high and low frequency information, they just came from two different sources.  This can be used as a way to combine the accurate high frequency information of a rate gyro with the accurate low frequency information from an accelerometer for attitude estimation.