Most jazz since the bebop era is based on a form that is actually quite similar to the sonata allegro form from classical theory: an optional introduction, the exposition or theme (possibly repeated), the development section, and the recapitulation, possibly followed by a coda. The introduction, if present, sets the tone for the piece; the exposition is the main melody; the development section is where the composer extends the ideas of the exposition; the recapitulation is a restatement of the theme; and the coda is an ending. In jazz terms, these sections of a piece would be called the the intro, the head (possibly repeated), the solo section, the head out, and possibly a coda or tag ending. The intro establishes the mood; the head is the main melody; the solo section is where the soloists improvise on the melody and/or chord progression of the tune; the head out is a restatement of the theme; and the coda or tag is an ending.
While not every piece follows this form, the vast majority of traditional jazz stays very close to it. During the solo section, the rhythm section generally keeps following the chord progression of the head while the soloists take turns improvising. Each time through the progression is called a chorus, and each soloist may take several choruses. In this respect, the theme-and-variations form of classical music is also a valid analogy. Each soloist plays an improvised variation on the theme.
The improvisation is the most important aspect of jazz, just as the development is often considered the most important part of the classical sonata. While listening to a piece, try to sing the theme to yourself behind the solos. You may notice that some soloists, particularly Thelonious Monk and Wayne Shorter, often base their solos on the melodic theme as much as on the chord progression. You will also notice that liberties are often taken with the theme itself; players such as Miles Davis, Coleman Hawkins, Sonny Rollins, and John Coltrane were especially adept at making personal statements even while just playing the head.
There are two very common forms for a head or theme in jazz. The first is the blues form, which is normally a twelve bar form. There are many variants on blues chord progressions, but most are based on the idea of three four bar phrases. In its original form, the second phrase would be a repeat of the first, and the third would be an answer to that phrase, although this convention is rarely adhered to in jazz. You may wish to check out the blues progressions listed later to get an idea of what they sound like, so you can recognize blues forms when you hear them. Liner notes and song titles will also often help identify which tunes are based on the blues. Some well known jazz tunes based on blues progressions include "Now's The Time" and "Billie's Bounce" by Charlie Parker, "Straight, No Chaser" and "Blue Monk" by Thelonious Monk, and "Freddie Freeloader" and "All Blues" by Miles Davis.
The other common form in jazz is the AABA song form, used extensively in popular music from the turn of the century until the dawn of rock and roll. This form consists two sections, called the verse or A-section and the bridge. The form is verse 1, verse 2, bridge, verse 3. The verses are similar or identical except for the lyrics and perhaps the last two bars. The song "I Got Rhythm" by George Gershwin, is one example of an AABA form. There are literally hundreds of tunes based on the chord progression to that tune, including "Anthropology" by Charlie Parker and "Oleo" by Sonny Rollins. Other songs with the AABA form include "Darn That Dream" by Jimmy Van Heusen, and "There Is No Greater Love" by Isham Jones. Songs such as these, popular songs from the first half of the century that have been interpreted by many jazz musicians, are often called standards.
These structures are only guidelines. Musicians such as Cecil Taylor showed us long ago that it is possible to express oneself without such well defined structures, and indeed this type of expression is often more personal that any more organized form. I have described these common structures to help you understand the context in which many musicians work, not to suggest that they are the only way. You should learn to discern for yourself when listening to other musicians what type of structures they are using, if any. You should also decide for yourself which structures to use in your own playing.