The above figure, called the Boring Figure, is the most famous of the "ambiguous illusions," which can be seen as two different figures, depending on how one looks at them. We'll tell you how ambiguous illusions work, and show you how the figure evolved.
The Boring Figure got its name from Edwin Boring, who wrote a paper about it in 1930. Boring was a psychologist interested in illusions and perception. He worked for some time with the Moon Illusion (the illusion that the moon is bigger when it is on the horizon than it is when it's high in the sky) and how people misperceive relative brightness. He liked the figure, commonly known as "My Wife and My Mother-In-Law," and wrote about it as an ideal ambiguous illusion.
Ambiguous illusions are illusions that are meant to shift from one object to another as a person's perception of them changes. A famous ambiguous illusion is the white-candlestick-two-black-silhouetted-faces illusion. Boring felt this one was too cut and dried, and promoted My Wife and My Mother-in-Law because he felt it was not as easy to put a line between where one "woman" started and the other ended.
This illusion has a venerable history. The first documented instance of it is in a German postcard from 1888. This is the most detailed version there is, with a lot of shading in the hair and eye-ear. This shading and detail work may be why the old woman seems to be the most prominent in the picture.
The next major publication of the figure came in 1915, in a humor magazine. This is when the image got the title "My Wife and My Mother-in-Law." William Ely Hill published the figure in the magazine with the title and the caption, "They are both in this picture. Find them!" Over the years the picture has gotten some variation, but has lost detail and shading. Now its most common form is the image at the top, which is a simple black-and-white drawing with no frills. (There is a male version, and this site you can see My Husband and My Father-in-Law.)
Ambiguous illusions are so powerful because even after we know that both images are on the page, we can only see one of them at a time. When we see the young girl, we can't simultaneously see the lines in context of the mother-in-law. When we switch our perception, the young girl disappears and we only see the old woman. Generally, we switch from one to the other by focusing on one part of the image. Looking at the cheek and nose in profile make the young girl spring forward. Looking at the "eye" will let us see the old woman. Essentially, when we focus in on certain details, our brain makes sense of the rest of the image around these contours. Ambiguous illusions show the way that vision is a work of both the eye and the mind. The eye takes in a set of lines, and, depending on what they are, the brain organizes them into a recognizable pattern which we then "see." We can't see both images at once, because, at least at first, it isn't possible for the brain to construct both images and overlay one on the other. While it is possible to train the mind into recognizing two sets of patterns at once, that fact that it is a process shows that "seeing" is still a matter of mental practice, not simply taking in an image passively with one's eyes.