FMCW Radar Signal Processing Flowchart / Ambiguity Function
The Ambiguity Function is just a name for 2D Correlator.  

It is known that given a shifted 1D Signal the optimal estimator for the shift (Range in RADAR / LIDAR, etc...) it the correlation with the signal.  

What if our signal had 2 parameters and both are shifted?
In the case of RADAR we have a shift of the signal due to the range (Time Shift) and due to velocity (Frequency Shift - Doppler Effect).

So how would you solve this?
Well, correlation with the transmitted signal shifted on both parameters.
The result of this operation is the Ambiguity Function which is a grid (Discrete) saying the correlation value between the received signal and a shifted (In both parameters) transmitted signal.