This article will have to be a little bit classical period biased to make the obvious, er... obvious, so bear with me. Binary Forms are common, and make for a great place to start learning to write basic forms.

Now, here is an important thing Theorists like to discuss:
1. A Form is Continuous if the cadence of the A section is something other than the Tonic of the Home Key.
2. A Form is Sectional if the cadence of the A section is on the Tonic of the Home key.

There are a couple of very common schemes that should be discussed:

| A | B |
then either, or both (more common) repeated:
||: A :||: B :||

The last example is called "Two Reprise", meaning the two "halves" get repeated. I've never heard anyone use "Single Reprise" for something like A A B or A B B, but I guess it's possible. Two Reprise is just by far the most common repeat pattern.

Now, from a compositional standpoint, what makes B B? B sections are delineated most often by either or both:
1. A change in melody
2. A modulation (or focus on some chord other than the Tonic, that may stand in for a different key).

In tonal music, both are common, though modulation is far more common in complex works (and both being used is common too).

Let's take a simple scheme with one key throughout.
The A section begins in the Tonic (Home) key (I or i).
The A section cadences in the Home key, or less commonly, with a Half Cadence (on the Dominant, V).
The B section begins in the Home Key (though often focuses on a chord other than I) and continues and cadences in the Home Key.

We could*call "Greensleeves" ("What Child is This" for the Christmas-spirited amongst you) a Sectional Binary Form:

A section of the common harmonization:
i - VII - VI - V
i - VII - VI-V - i
B section:
III - VII - i - V
III - VII - i-V - i

Now, if you know this piece, the melody is VERY similar throughout - only subtle differences for the syllables. What makes the melody have a B section is that it begins a little differently for the 3rd and 4th line - enough so though that we know it's a new section (*I say "could" call this Binary Form because really this is a Phrase-Level analysis I'm "upping" to Piece-Level for illustrative purposes).

The harmony is also a clue. Does it really modulate to the Relative Major for just two chords (III and VII as I and V of the relative Major)? Not really. So we stay in the Home key throughout. But it's the focus on new melodic patterns, and in this harmonization (because this piece is harmonized differently in other settings) the "relative majorish" chord beginning what we now call the B section.

A more obvious example is "Yankee Doodle" (also Sectional Binary):
A section:
C C D E C E D | C C D E C B
C C D E F E D C | B G A B C C

B section:
A B A G A B C | G A G F E G
A B A G A B C A | G C B D C C.

Not only is the B section obviously starting on the A note, but it includes a dotted rhythm we haven't heard before, so this is obviously "new" material compared to the opening material. The typical harmonization also focuses on the IV chord at the start of the B section (*again, this is a Phrase-Level piece but the analysis compares well to Piece-Level analysis).

Historically speaking, the far bigger indicator of a new section (and this is what many people don't understand about tonal music) is modulation to a different key. In fact, you can have the exact same melodic material and change the key, and we'll call that enough to make it a new section!

In other words, presentation of material in a new key is a far bigger indicator of a new section (even if the material is similar to or identical to the original material) than melody alone (or any other factor).

There are two ways to do this:
1. Close the A section in the Tonic of the Home Key (Sectional).
2. Modulate to the new key before closing the A section (Continuous).
3 And for either, the B section begins in the new key and modulates back to the Tonic of the Home Key for the final cadence (there are a very very few select pieces that never modulate back to the Tonic, and these are typically part of a larger set that serve as connectors, not stand alone pieces).

Now, here are the common modulation schemes:
1. If a piece begins in Major, it will most likely modulate to the key of the Dominant, or less likely, the relative Minor.
2. It a piece begins in minor, it will most likely modulate to the key of the Relative Major, or less likely, the Dominant.

Note that this means in pieces that don't modulate to a new key, Sectional means the A section ends on I, and Continuous means the A section ends on a half Cadence (V). In pieces that do modulate to a new key, Sectional still means the A section ends on I of the home key, but Continuous means the A section ends on the I or V of a NEW KEY (unlikely that it ends on V of the original key, but possible).


So usually, the formal schemes for "modulatory" forms are written like:

A: I ~ ~ V (vi) | B: V (vi) ~ ~ ~ I

or

A: i ~ ~ III (v) | B: III (v) ~ ~ ~ i

(both Continuous forms)

And it's understood that the ending of the A section might not modulate until we actually start the B section, so:

A: I ~ ~ ~ I | B: V (vi) ~ ~ ~ I
(Sectional form)

You just don't always see both options written, but they're both possible.

What you should do as a composer if you're into exploring these things, is try to write one of each:

"Simple" Binary - Remain in one key throughout, and let your Melodic material indicate the A and B sections, or let your your Harmonic and Melodic material indicate the A and B sections. Try an example that's Sectional (A section ends on I of the home key) and try an example that's Continuous (A section ends on V).

"Complex" Binary - Modulate to the keys in the schemes above. You could try a simple version of this where the melody doesn't change, just transposes (I know of a few pieces like this, one by Rameau in particular) or go the more typical route and make the melody change with the modulations.

One of the best sources for examples of these types of Binary Forms are from the little dances by and collected by Bach - things from the Anna Magdelena Notebook (Minuets, Bourees, etc.) and the like - a lot of Dance forms.

Here's an interesting little test you can take:
http://www.learn-ict.org.uk/resource...usic/form2.htm

And here's a wonderful resource:
http://www.d.umn.edu/~jrubin1/JHR%20Binary3.htm

Something the latter site points out that's worth mentioning - Your A and B sections don't have to be equal length. Many times one is longer than the other, often the B section.

Once you get a handle on these more familiar types, you can explore other ways of delineating A and B sections, or other schemes - A section in Dorian, B section in Lydian and so on.

You don't have to stick to the classical model. Try to see how many different ways you can make two different sections work together. Try to see how similar you can make them while maintaining two sections, or, try to see how different you can make them before they don't sound like they belong together anymore.

fine.