The contrast equalizer is designed to tune the amount of local contrast within the image.
Selectively boost local contrast (luma tab), or saturation (chroma tab) around reduced areas, following those two axis:
Horizontally: coarse -> fine
Vertically: contrasty -> smooth
Move the mix slider below to change the amount of the effect that is applied.
Luma tab
The luma tab is about adjusting local contrast.
When I want to strongly boost the overall contrast in a cleanly distributed way, I personally tend to move the spline up above the first cursor from the left. It can be moved down in the opposite direction to soften the image.
Move the cursor past the middle of the histogram to increase fine details.
Chroma tab
The chroma tab will adjust the saturation in a more or less subtle way. As it is very difficult to quantify the best way to see how it will affect the picture is to experiment with it.
When to use: the contrast equalizer is a very powerful tool, use the luma tab when you want to stylize the image in a very contrasted way or on the contrary soft way.
The chroma tab is mostly affecting color saturation, but raising it up can also reveal details in colorful and detailed elements such as flowers or trees.