This article is mainly addressed to those new to photography, although may be useful for more advanced photographers.
There are plenty of articles written on curves today, but at the same time not enough. Most of these articles are about Curves tool in Adobe Photoshop. But curves themselves are much broader and important term. Although it is really easier to study how they work in Photoshop, it should be comprehended that curves are integrated in almost any photography software, one way or another. Even if you do not see familiar instrument, it probably exists and affects image. Even when you adjust Brightness or Contrast in Lightroom, you are actually control some kind of curve. Not to mention displaying picture on your screen is impossible without gamma-adjustment and managing brightness-contrast characteristics through display ICC profile, which are also controlled by curve. Moreover, when you shoot on film, image which you see after developing is affected by characteristic curve of particular film you have used.
Thus, despite what kind of camera you use and how much attention you pay to developing image in either dark or light room, understanding curves is rather crucial. And not only for photographers: scanning clerks, color-correctors, designers, print workers and many other specialists who work with raster graphics use curves. Hence, curves is primary (often the only one) and most effective tool to process pictures.
One can write a whole book on curves, and I will not be surprised if it already exists. This article is by no means exclusive or, moreover, comprehensive. In fact, my laziness made me write it – it’s easier for me to write one article than to explain the same thing to students every time. Articles I have read do not serve my teaching purpose, for one reason or another. Another reason why I wrote it is that none of the articles I happened to read presented an adequate guide to Curves. Most of them are too focused on Curves in Photoshop, not elaborating enough on key relationships between curves shape and its effect on picture.