Dangerous Curves Ahead
Curves are a tool for adjusting tonal range, common to many image editing applications. Some programs make Curves easier to use than other. Others give you greater control, but in so doing, leave you open to damaging your image unless you know what you are doing.
First, a brief overview of how Curves work. The basic design of the Curves adjustment tool is the same in pretty much every application that has such a thing. There's a basic grid or graph with the bottom axis representing brightness of the unaltered image, from pure black on the far left to pure white on the far right. The vertical axis on this grid likewise has black at the bottom and white at the top, and represents the tonal range of the image as modified by the Curves adjustment. Overlaying this grid is a line that maps input brightness along the bottom axis to output brightness along the vertical. This line starts out straight, from the lower left-hand corner to the upper right, mapping each input value to the same value as output. Such a straight line may not look much like a curve, but that's because it isn't yet doing anything to modify the image. Adjusting an image with Curves consists of pushing, pulling and dragging that initial line to change its shape, generally making it look more like a curve. In so doing, you change the relationship of input values to output, altering the tonality of the image.
Just like with Levels, Curves doesn't directly tell you where any given tonality can be found in an image. It acts on all parts that have the same brightness equally. Curves modifies and rearranges contrast globally. If you want it to affect only a portion of an image, you would need to use a layer mask or some other means of limiting the effect.
When you first try to adjust an image using Curves, the task can seem a tad overwhelming. You're faced with a straight line leaning up and to the right, with no curve to be seen anywhere. It's not even apparent why the call the tool "Curves." Lightroom makes Curves relatively safe, restricting you to editing just the highlights, or just the shadows. When you go to move the curve line, Lightroom corrals you into limiting your adjustment to a range that won't mess things up too badly. But Photoshop lets you do whatever you want with that line, even when you probably shouldn't. As you manipulate that line into becoming more curve-like in Photoshop, it's easy mess an image up so badly that you never want to venture into Curves ever again. But once you know how to use it, Curves can be one of the best tools available to improve an image. Really.
Using Curves effectively can take some practice. But to avoid disaster, there are two main problems you need to avoid.
One major problem has to do with the slope of the Curve line. It starts out leaning upwards towards the right, and it needs to remain leaning upwards. You can push and pull all you want, but you should never drag it down so low that it slopes downhill as it moves right. Negative slope is bad. Since each point along the horizontal axis starts off brighter than every point to its left, it would be expected that it maps to a brighter output value along the vertical axis too. If, instead, it maps to a darker value, things start to look weird. What starts out dark in your image will appear light. And what starts out light will end up dark. Your image will look somewhat like a film negative, if you're old enough to know what that is.
The other problem relates to how sloping uphill too quickly. Especially if you are working in 8-bit mode, there are a limited number of possible values any given pixel can have. If you slope the curve line upward too drastically, you will end up with gaps in the output values significant enough to cause banding and similar visual problems in an image. Normally, 1 maps to 1, 2 to 2, and so on. But if you make the curve too steep, 1 may still map to 1, but now 2 perhaps maps to 10, and 3 to 20. Miss enough values on the output axis, and you will be sure to notice, especially in areas such as the sky where smooth gradients are expected.
For completeness, I should note that CMYK images usually display these axes reversed in the Curves dialog, with black on the right and white on the left, but most of us photographers work in RGB, not CMYK. If you are one of those pre-press folks who do work in CMYK, the same issues remain, just with the directions reversed.
Curves can be powerful, but they can also be dangerous. As a general rule when making Curves adjustments, a little goes a long way. Anything too drastic is probably a bad idea.