A texture describes the way that melodies behave and interact with each other in a piece. There are three textures that account for the majority of western music: monophony, polyphony, and homophony.
When a piece of music is nothing more than a melody, this texture is called monophony. There may be only one voice or instrument (monophony literally means "one sounding") or there may be several, all performing the same line of music. When all the performers are playing or singing the same notes, they are performing in unison. When a room full of people sings "Happy Birthday", the men are usually singing the melody an octave lower than the women, so they are no longer singing in unison but at the octave. However, due to octave equivalence (see the section on pitch), the same melody sung in a different octave still sounds the same, so singing at the octave is also considered monophony.
The body of Medieval religious chant known as "Gregorian" chant is largely monophonic.
Polyphony, meaning "many sounding", refers to music in which several independent melodies occur simultaneously. These melodies intermingle and overlap, and are generally designed to sound good together.
Another term that is closely related to polyphony is counterpoint. Counterpoint is a style and method of writing polyphony that was used from the 15th through 17th centuries. So, a discussion of counterpoint is really a discussion of a specific type of polyphony. In many cases the two terms can be used like synonyms (example: contrapuntal texture = polyphonic texture).
There are two subtypes of polyphony: imitative and free. Imitative polyphony occurs when the melody of the first voice is copied by subsequent voices. The most strict form of this type of imitation is the canon. In a canon, the original melody is emulated precisely and without variant in every voice. Perhaps the most well-known canon is "Row, Row, Row Your Boat": (if the image is too small, click to view it larger)
"Row, Row, Row Your Boat"
A fugue is another imitative polyphonic form. Fugues are less strict than canons: the different voices begin by imitating each other, but gradually diverge and become unique. Bach's "Little" G Minor Fugue is an example of this type of imitative polyphony. The video below allows you to follow the basic contours of the different parts without needing music notation. For more on the fugue, see the section on form in popular and art music.
J.S. Bach, "Little" G Minor Fugue
Free polyphony, on the other hand, can be found in traditional New Orleans jazz and in the early polyphony of the late Medieval and early Renaissance periods. In this texture, the independent voices are each unique and do not copy each other. "Hotter Than That" performed by Louis Armstrong and His Hot Five, is an example of New Orleans jazz that begins with a section of free polyphony followed by improvised solos (trumpet, clarinet, voice, then trombone).
"Hotter Than That" performed by Louis Armstrong and His Hot Five
When there is one melody, and the other parts are designed to support and accompany the melody (the accompaniment), then the texture is homophony. The majority of western music falls into this category. For our purposes, any music that is not monophonic and not polyphonic can be considered homophonic.
There are two main types of homophony: melody-and-accompaniment, and chorale-type (homorhythmic) homophony. The majority of popular music, art song, and opera falls into the first category. To be considered chorale-type homophony, all the voices (or instruments) must be homorhythmic - all executing the same rhythm at the same time. The different parts will be singing different notes (otherwise it would be monophony), but they will say the lyrics at the same time and move with the same rhythm.
The beginning of Queen's "Bohemian Rhapsody" is a good example of chorale-type homophony. The rest of the song is predominantly the melody-and-accompaniment type of homophony.