Binary form

Binary form is a musical form in 2 related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance. In music this is usually performed as A-A-B-B.
Binary form was popular during the Baroque period, often used to structure movements of keyboard sonatas. It was also used for short, one-movement works. Around the middle of the 18th century, the form largely fell from use as the principal design of entire movements as sonata form and organic development gained prominence. When it is found in later works, it usually takes the form of the theme in a set of variations, or the Minuet, Scherzo, or Trio sections of a "minuet and trio" or "scherzo and trio" movement in a sonata, symphony, etc. Many larger forms incorporate binary structures, and many more complicated forms (such as the 18th-century sonata form) share certain characteristics with binary form.
Structure.
A typical example of a piece in binary form has two large sections of roughly equal duration. The first will begin in a certain key, which will often, (but not always), modulate to a closely related key. Pieces in a major key will usually modulate to the dominant, (the fifth scale degree above the tonic). Pieces in a minor key will generally modulate to the relative major key, (the key of the third scale degree above the minor tonic), or to the dominant minor. A piece in minor may also stay in the original key at the end of the first section, closing with an imperfect cadence.
The second section of the piece begins in the newly established key, where it remains for an indefinite period of time. After some harmonic activity, the piece will eventually modulate back to its original key before ending. 
More often than not, especially in 18th-century compositions, the A and B sections are separated by double bars with repeat signs, meaning both sections were to be repeated.
Binary form is usually characterized as having the form "AB", though since both sections repeat, a more accurate description would be "AABB". Others, however, prefer to use the label "AA′". This second designation points to the fact that there is no great change in character between the two sections. The rhythms and melodic material used will generally be closely related in each section, and if the piece is written for a musical ensemble, the instrumentation will generally be the same. This is in contrast to the use of verse-chorus form in popular music—the contrast between the two sections is primarily one of the keys used.
Further distinctions.
A piece in binary form can be further classified according to a number of characteristics:
Simple vs. rounded.
Occasionally, the "B" section will end with a "return" of the opening material from the "A" section. This is referred to as rounded binary, and is labeled as "ABA"′. In rounded binary, the beginning of the "B" section is sometimes referred to as the "bridge", and will usually conclude with a half cadence in the original key. Rounded binary is not to be confused with ternary form, also labeled "ABA"—the difference being that, in ternary form, the "B" section contrasts completely with the "A" material as in, for example, a minuet and trio. Another important difference between the rounded and ternary form is that in rounded binary, when the "A" section returns, it will typically contain only half of the full "A" section, whereas ternary form will end with the full "A" section.
Sometimes, as in the keyboard sonatas of Domenico Scarlatti, the return of the "A" theme may include much of the original "A" section in the tonic key, so much so that some of his sonatas can be regarded as precursors of sonata form. 
Rounded binary form is sometimes referred to as small ternary form.
Rounded binary or minuet form:
 A :||: B A or A'
 I(->V) :||: V(or other closely related) I
If the "B" section lacks such a return of the opening "A" material, the piece is said to be in simple binary.
Simple:
 A->B :||: A->B
 I->V :||: V->I
Slow-movement form:
 A' A"
 I->V I->I
Many examples of rounded binary are found among the church sonatas of Vivaldi including his Sonata No. 1 for Cello and Continuo, First Movement, while certain Baroque composers such as Bach and Handel used the form rarely.
Sectional vs. continuous.
. Note: the example here is in minor mode rather than the more historically accurate Dorian mode.
If the "A" section ends with an Authentic (or Perfect) cadence in the original tonic key of the piece, the design is referred to as a sectional binary. This refers to the fact that the piece is in different tonal sections, each beginning in their own respective keys.
If the "A" section ends with any other kind of cadence, the design is referred to as a continuous binary. This refers to the fact that the "B" section will "continue on" with the new key established by the cadence at the end of "A".
Symmetrical vs. asymmetrical.
If the "A" and "B" sections are roughly equal in length, the design is referred to as symmetrical.
If the "A" and "B" sections are of unequal length, the design is referred to as asymmetrical. In such cases, the "B" section is usually substantially longer than the "A" section.
The asymmetrical binary form becomes more common than the symmetrical type from about the time of Beethoven, and is almost routine in the main sections of Minuet and Trio or Scherzo and Trio movements in works from this period. In such cases, occasionally only the first section of the binary structure is marked to be repeated.
Although most of Chopin's nocturnes are in an overall ternary form, quite often the individual sections (either the A, the B, or both) are in binary form, most often of the asymmetrical variety. If a section of this binary structure is repeated, in this case it is written out again in full, usually considerably varied, rather than enclosed between repeat signs.
Balanced binary.
Balanced binary is when the end of the first section and the end of the second section have analogous material and are
organized in a parallel way.