As you may know from experience, as you slow down a tape, the pitch drops. Of course only the delayed copy of the sound has this pitch change, which is then mixed in with the unaltered signal. This is the depth parameter referred to above. The larger the depth, the more pronounced the notches in the flanger. In multieffects units, the depth may only be controllable in the mixer section, and not available within the flanging processor.
Some people use the term 'mix' interchangeably with 'depth'.
The delay parameter specifies the minimum delay used on the copy of the input signal - as the delay changes, this will be the smallest delay. Looking at the frequency response, this value determines how high the first notch will go. As the delay is increased, the first notch drops down. In some cases, the delay parameter can be set to zero, in which case the notches will sweep the uppermost frequency range, and essentially disappear momentarily. In other cases, you may not be able to control delay parameter.