Borromean rings are three rings which are interlocked in such a way that removing one ring causes the entire structure to fall apart. This is an illustration of what is known as a Brunnian link, a situation where no two loops in a figure are directly connected. In addition to being represented with circles, Borromean rings can also be made with a variety of other shapes, or by combining several shapes together. Oddly enough, when Borromean rings are constructed in three dimensions, they cannot actually be made with three flat circles.
As an artistic motif, Borromean rings are extremely ancient. They appear in Buddhist art from thousands of years ago, for example, and they can be seen on Viking rune stones, in Roman mosaics, and in an assortment of other places. People appear to have an enduring fascination with the phenomenon of Borromean rings, and they appear especially frequently in religious artwork from a variety of cultures. In Christian art, for example, Borromean rings are often used to represent the Holy Trinity.
When used in religious artwork, coats of arms, logos, and crests, Borromean rings are meant to symbolize strength in unity, a living illustration of what happens when one link in a united element is removed. The rings are named for the Borromeo family of Italian nobles, who famously used them in their family coat of arms, popularizing the three interlocked rings.
Using three dimensional modeling, mathematicians have shown that Borromean rings cannot be produced in real life, because the rings would need to be distorted to create the desired Brunnian link. However, they can be made from ellipses, triangles, squares, and other shapes. You may have seen Borromean rings in the form of toys which require people to remove one link, causing the toy to fall apart. Stores which specialize in logic puzzles and brain teasers often stock Borromean rings.
Borromean rings are only the most simple form of the Brunnian link. Brunnian links can be made with a potentially infinite number of shapes, and they can get extremely complex and actually rather pretty. Members of the scientific community have studied Brunnian links to see how they apply to things like the structure of molecules, and some early attempts at modeling the shape of DNA were made with Brunnian links.