In our usage cross-correlation is a generalization of autocorrelation to a pair of time series:
C(T)=E[x(t),y(t+T)]|over all available t
where T is the time lag. We will also use the more context-specific term intermarket correlation for certain cross-correlations. Just like significant autocorrelation at non-zero time lags lets one predict (in the statistical sense) the process on the basis of its own past history (“auto-” means “self-“), significant cross-correlation at non-zero time lags lets one predict X on the basis of Y (in the same sense with the same caveates) or Y on the basis of X, depending on the sign of T where the signal occurs. For example if you notice that XXX/YYY lags begind UUU/VVV, and you know that UUU/VVV just went up, you can go long on XXX/YYY and have a better than average chance of ending up with a winning trade.
Feel free to browse our collection of cross-correlation data for the forex markets — some of them are highly non-trivial.