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Curves is a very useful tool in Photoshop used to adjust the tonal range in an image. It is a more sophisticated and precise tool than the Levels command. Where Levels can change shadows, highlights, or midtones, Curves can change up to 16 values at once independent of each other. It is a very powerful tool though it can be a little intimidating when you first look at it.
 Prior to editing the image using curves, it is a good idea to convert the image to Lab Color mode (Image --> Mode --> Lab Color). This will ensure that any image posterization and color shifts are kept at a minimum. Once you are done with the Curves adjustments, you can convert the image back to the previous mode.

The Curves dialog box is opened by selecting Image --> Adjustments --> Curves. What you get will look like this.
Curves Dialog 
Now this seems a little confusing so let me explain what does what.

At the top you have a certain number of presets for conditions such as high and low contrast, negative, color negative etc. You can also create your own custom presets.

The channel should be set to Lightness. If you were doing these changes in RGB color then the Channel would say RGB.

The horizontal axis of the graph represents the original pixel brightness values called Input Levels. The vertical axis represents the edited brightness values called Output Levels.

When you initially open the Curves dialog, the line is in the default diagonal position. This means that each pixel has the same Input and Output value. In Lab Color, the curve shows brightness as a percentage, with the brightest in the upper right corner and darkest in the lower left corner. If you were in RGB mode, this would be a range of 0 to 255.
Editing your image using Curves is fairly easy. First thing to do is to create some control points. This is done by simply clicking on the diagonal line where you want to create one. This can be anywhere on the diagonal. It does not have to be as you see it below. You will generally only need 3 but you can create more if you need to.
Curves Control Points
Now comes the fun part. Moving control points will change the tonality of your image. Dragging highlight control points downward will darken the image highlights, while dragging it upwards will brighten them. The same thing is true for shadow control points, but they only affect shadows. You can also adjust the midtones by using a control point near the middle of the diagonal.
Adjusted Curves
This type of s-shape would decrease the contrast in a image. An inverted s-shape would increase it (shadows dragged downwards and highlights upwards).

You can see why this is a much more precise tool than Levels. You can adjust multiple data points along the diagonal independently of each other. The image above is very similar to what you would do using the Levels Adjustment. One shadow, one highlight, and one midtone control point is all you get when you use Levels. Using Curves, you can set quite a few points to achieve the effect you want.
Multiple Control Points
Please be aware that the above adjustment creates a very radical looking image, I only used to show how multiple control points can be used.

Dragging all control points downward will darken the entire image. Dragging them all upwards will lighten it.

Curves is a very powerful tool once you understand how to use it. It offers a lot more control over Levels and is much more precise. Keep in mind that all editing should be done in Lab Color to limit posterization and color shifts, and converted back to your original mode when you are done with Curves. Also remember that is is much better to make tonality changes in 16-bit mode. Have fun.

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