The terms monophony and polyphony have very straight-forward literal meanings. Monophony means music with a single "part" and a "part" typically means a single vocal melody, but it could mean a single melody on an instrument of one kind or another. Polyphony means music with more than one part, and so this indicates simultaneous notes.
In practice, these simple definitions can be blurred by various performance techniques or refined by other terms. The principal example of monophony is plainchant, with its single unaccompanied vocal melody. When sung by multiple voices in unison (i.e. the same pitch), this music is still considered monophonic. When doubled at the octave or other interval, as is done not infrequently in practice, it is arguably homophonic (see below). However, this is the sort of precision which only plainchant specialists would typically consider important. Another important example of monophonic music is the troubadour repertory. Although the surviving musical sources are monophonic, they are often performed with accompaniment today. Literally speaking, this would make them monody in practice (see below). Finally, the music of e.g. Hildegard is also monophonic, and of course closely related to plainchant.
Although polyphony literally means more than one sound, and so any example of non-unison doubling or accompaniment would be polyphony in the strict denotational sense, the word generally has a more specific connotation. Namely, it suggests that there is melodic interest in each part, and rhythmic distinction between each part. It frequently implies even rhythmic independence. Homophony, in contrast, implies no such independence. In homophonic music, multiple parts generally move in the same rhythm. This could also be called chordal music. One could therefore suggest that early note-against-note organum is homophonic, but the word is not generally used in that context.
There is often some confusion between the terms monophony and monody. Monody is a term with a definite historical origin. The 16th century madrigal was a polyphonic secular song form, with melodic interest shared between the (most frequently 5) voices. In the development of the more soloistic style which was one of the driving forces in the origin of the Baroque, and with it modern tonality, emphasis was shifted to a single upper line for melodic interest as accompanied by instrumental parts to fill a harmonic texture. In a prototypical example, the latter could be chords on a lute. Monody was the name given to this style. From this perspective, one might note that even recent orchestral music is frequently monodic: i.e., a primary melody in the upper range accompanied harmonically. There is some lingering overlap between the terms homophony and monody. The term monody emphasizes the distinct or soloistic role of the main melody, while the term homophony emphasizes the concord and alignment between voices in the texture. In practice, it may be difficult to give many sections of "common practice" music one label or the other. The quodlibet is frequently in quintessential homophonic form, as is the later "barber shop" music.
Another term which sometimes appears is heterophony. Heterophony means that multiple parts use the same melody, but at somewhat different times. In other words, it is like doubling, but not at the same time. The term heterophony was invented to distinguish many world musical styles from Western polyphony, and so is sometimes considered prejudicial. It does, however, designate a more specific kind of polyphony. In heterophony, generally speaking, any vertical alignment of intervals is coincidental and not important. This is as distinguished from a fugue or other imitative forms, which we might otherwise term heterophonic.
Finally, some discussion of when these words appeared in the English language may shed further light on the nuances of their meanings. Monody appeared in print in 1589, as part of the original discussion of this music when it was new. Homophony appeared with Burney in 1776, emphasizing the concord of harmonized melody. Polyphony appeared in 1864 to distinguish certain contrapuntal sections from homophony. Monophony appeared in 1890, as the clear analog to polyphony. Heterophony finally appeared in 1919, as a term to apply to music of other cultures, as noted. This sequence illustrates that the concept of accompaniment has always been central to monodic music, and that monophonic music per se was distinguished only later.
Addendum on contrapuntal or counterpoint: When this word first appeared in (Latin) theory around 1300, it designated note-against-note writing which we might call homophonic today (I emphasize "might" because this style usually had melodic interest in each part, rather than a main line and accompaniment) as opposed to what was then the more typical polyphonic style. Today, the meaning of contrapuntal is virtually reversed: It indicates a polyphonic texture, often with rhythmic independence. This is only one of the ironies which can continue to cause confusion in the use of these terms for music of different eras.
To Early Music FAQTodd M. McComb