Modern Western Square Dancing is a form of folk dancing with roots in Morris dance, English country dance, the French Quadrille, traditional Appalachian Mountain dance, New England country dance, and other American reels and squares. The term square dancing derives from the formation of the dancers: four couples arrayed on each side of an imaginary square. Before the 1940s and 1950s, square dances were more highly formalized; the first couple would move to the couple to their left, complete a sequences of dance steps and movements with them, and then move to the next couple in sequence, repeating the same steps. As couple one moved to couple four, couple two would then move to couple three, and so on, until every couple had completed the sequence with every other. These types of traditional squares are often still danced between contra sets at contemporary contra dances.
The 1940s and 1950s, however, began to see the development of a more extemporaneous form of square dancing, the movements of which have continued to be codified into what is now formally known as modern western (or western-style) square dancing, but usually called just square dancing.
A square dance (generally) requires four couples and a caller; the caller recites or sings series of movements, which the couples attempt to execute. Except in certain stylized dances like those described above as traditional square dances, the dancers do not know which movements they will be asked to do until the moment the caller speaks them. These movements within square dancing are referred to as "calls"; calls have specific definitions, determined and recorded by the group Callerlab, an international association of square dance callers. Moreover, these calls are grouped into several formalized square dance levels, beginning with Basic and Mainstream, followed by Plus, Advanced 1 and Advanced 2, and ending with Challenge 1, 2, 3a, 3b, and 4. While callers in non-English speaking countries learn the definitions of calls in their own languages, the calls themselves are always the same (most are English phrases, though some, like allemande or dosado, the two best-known, are squaredance terms that derive from the languages used in earlier dances). The benefit of this as well as of the organization of calls into levels means that anyone who has mastered a particular level can go anywhere in the world and dance a square dance advertised as being at that level.
At each level, the dancers are required to memorize the definition and execution of more calls, new formations (lines, columns, circles, figures that look like the letter H, the letter I, or a stylized butterfly, for example), and new concepts; concepts are not themselves sequences of movements, but they modify the way calls are interpreted and executed by the dancers (much the same way as an adverb modifies a verb). For example, the Challenge concept called "tandem" requires that dancers form a single unit with the person immediately in front or behind them, and perform the next call as a unit rather than as individual dancers. Another concept, "split," requires that the dancers think of the formation as split into two smaller formations, and to perform the call only within their half, not crossing over an imaginary dividing line between the two, rather than within the larger whole.
At the Advanced and Challenge levels, square dancing bears very little resemblance to the activities that many of us were required to perform in elementary school, and that we usually think of when we hear the phrase "square dancing." Dancers at these levels have memorized several hundred to more than a thousand separate calls and definitions, and a couple dozen formations and concepts, any of which they must execute on the spot. They must also be able to break calls into their component parts so that they may fractionalize a call by x/n — that is, do only the first x parts of the call's n total parts — or interrupt one call at a certain point, execute another call, and then complete the remaining parts of the first call; or execute a sequence of moves with one or more "phantom" dancers — imaginary dancers occupying the relevant empty spaces in your formation. Challenge dancing, especially, is more an intellectual, logical, puzzle-solving activity than a form of physical exercise or purely social dance. Both MIT and Stanford University boast very active and highly regarded square dance clubs.
Basic and Mainstream - 68 calls/concepts
Plus - 32 additional calls/concepts
Advanced 1 - 48 additional calls/concepts
Advanced 2 - 37 additional calls/concepts
Challenge 1 - more than 70 additional calls/concepts
Challenge 2 - more than 70 additional calls/concepts
Challenge 3a - more than 80 additional calls/concepts
Challenge 3b - more than 70 additional calls/concepts
Challenge 4 - hundreds of additional calls/concepts
A sample square dance call definition:
From Parallel Two-Faced Lines.
Leaders Any Shoulder Turn & Deal as Trailers 1/2 Circulate; Very Centers Hinge; Flip The Diamond.
Ends in a 1/4 Tag.
The definition describes the permissible starting formation(s), in this case parallel two-faced lines; gives the component steps, for which this has three; and notes the ending formation(s) (sometimes the ending formation changes depending on which of several starting formations was used).