Applications of Correlation in Signal Processing
A relatively simple application where correlation is used is in determining timing of a communications signal.  A known synchronization signal will be sent periodically so that the receiver can use it as a point of reference.  How does the receiver use it as a point of reference?  It correlates the incoming signal stream against the known sequence, and when the correlation peak is above a certain threshold, it will use the peak to establish symbol timing and make decisions on the symbols to extract a binary sequence.

Correlation usually means cross-correlation, but I think the latter term only exists to be distinguished from auto-correlation.