Reading the graph
The horizontal axis is the tone a pixel has now (input); the vertical is the tone it will have (output). The default diagonal means "leave everything alone". Drag a point up and those tones get brighter; drag down and they get darker. Shadows sit at the bottom left, highlights at the top right.
The S-curve
Add contrast by placing one point in the shadows and pulling it down slightly, then one in the highlights pulled up slightly. This is an S-curve, and it is the single most useful move in the dialog. Steepening the middle increases contrast there while compressing the extremes — which is exactly what film did and why the result reads as natural.
Set black and white points first
Before any shaping, drag the bottom-left endpoint right until it meets the first real data in the histogram, and the top-right endpoint left to meet the last. Hold Alt while dragging to see exactly which pixels are clipping. Most flat-looking photos are flat because these points were never set.
Fixing a colour cast
Switch the channel dropdown from RGB to Red, Green or Blue and you are editing colour, not brightness. Raising the Blue curve adds blue; lowering it adds yellow. Red against cyan, green against magenta. A photo shot under tungsten light usually needs blue up and red down slightly — see fixing white balance.
Keep it local
A Curves adjustment layer arrives with a mask, so any curve can be confined to part of the picture. Add the curve, invert the mask to black, and paint the effect exactly where you want it. Most strong edits are several small local curves rather than one global one.
The targeted adjustment tool
The little hand icon at the top left of the Curves panel lets you click a spot in the image and drag up or down to move the exact part of the curve controlling that tone. It is the fastest way to brighten "that bit there" without hunting along the line.