The sonata form dominated musical structure from the Classical Period all the way through the 20th century. It grew out of the rounded binary form of the early Classical Period. Any discussion of sonata form will involve at least some basic harmonic terms, so we advise you visit Lesson 8 of our Music Theory Section if you find yourself getting confused at any point.
Although almost every chamber work from the Classical and Romantic periods contains at least one movement in sonata form, that does not mean that composers had a set of rules that they followed to create works in that form. Rather, the "rules" of sonata form were later determined in hindsight by studying the works that had been written and finding similarities between them. For these early composers, the sonata form was just a natural form of musical expression.
The concept, while it can be made very complicated, is actually very simple. First, you introduce the musical material that will be explored in the movement. Then, when the audience has become accustomed to your material, you change it around and develop it in interesting ways. Finally, you bring back the original material, giving a sense of closure to the whole thing. That is the easy definition.
Let's expand on that concept a little... the opening section where we introduce all our material is called the exposition. This section is always in the home key (the tonic). As the music is introduced, the music undergoes a modulation and arrives at the end of the exposition in a different key, almost always the dominant.
Now that the musical material has been established and the harmony has moved off into a different key, the movement moves into the development stage. This part is generally free-form, and the composer is largely free to do whatever he wants. Different composers will do different things in their developments. Some like to move the music through several keys, creating a sense of unrest; others may stress the contrasts between the themes from the exposition, creating a tension within the work. No matter what the composer does, however, the one thing that must happen ion the devlopment section is that the music must somehow return to the home key.
When the music returns to the tonic key, we have arrived at the recapitulation. Think of this as a "recap" of all the themes that we have seen during the course of the movement. This section restates the material from the exposition, but this time the music does not move to a different key. Having everything retold entirely in the home key gives a sense of closure to the entire piece, as all harmonic tension has been relieved.
Sometimes sonata form movements will add a slow introduction
and/or a coda. The function of an introduction is to set a more
serious tone or create the impression of a larger scale. Also, the
introduction may also have been a signal to the audience that the
music was about to begin. A coda is a section of music that is
added onto the end of a piece; it is usually used to restate the
main theme(s), creating even more of a sense of closure than the
recapitulation alone can provide.