We see the concept of counterpoint in many contexts, from classical music composition to jazz improvisation.
What is the definition of counterpoint?
Counterpoint comes from "punctus contra punctus" - point against point. This stems from the notion of having a given melody, the "cantus firmus" against one writes a counter melody. The most prominent place for this type of counterpoint is in courses on counterpoint.
More generally, counterpoint simply means the technique of writing multiple independent melodies that sound at the same time - i.e. polyphony.
While polyphony literally means: many sounds, this is different from melody with harmonic accompaniment. The difference is that in that case, there is one prominent melody (salient line) which is merely supported by a subordinate number of voices which are not really independent but simply provide a background. This may and often does consist of multiple different sounding notes (chords) but is still called homophony.
So to call something counterpoint, the voices have to be "independent". This is a subjective concept but in general the sense of independence is strengthened when the melodies have different rhythms (for example by syncopation), and also when the melodic movement display different motion (contrary motion, i.e ascending contours are placed against descending contours).
Those are just basic mechanisms of independence, in real counterpoint one will subjectively recognize the melodies as being separate entities, i.e. each melody has separatately recognizable themes and motives (and of course, this is achieved at some point by employing different rhythms and melodic curves).
That said, this does not mean that polyphony does not have harmony and that all homophony is entirely un-contrapunctual. The best kind of counterpoint is when the melodies are truly independent, and also agree in harmonic sense by articulating a clear progression of chords.
Education in counterpoint is often based on or at least refers to Fux' "Gradus ad Parnassum". There is an english translation available here: http://www.opus28.co.uk/Fux_Gradus.pdf
The title refers to Mount Parnassus, which symbolizes a summit of mastership, and the "gradus" are steps, as in a stairs. The idea of this book is to gradually guide a student trhough different "species" of counterpoint. The first part of the book is dedicated to 2-part counterpoint, where the student writes against a given melody (cantus firmus). In the first species, the student can only write one note for each note in the given melody, in the second species two notes against each given note, and so on and so forth.
In the second part of the book this technique is applied to 3 and eventually 4 part writing. However the weakness of Fux' approach is that it has no notion or regard for the harmonic facets of writing counterpoint (and in my opinion, this facet becomes more important as more and more voices enter the counterpoint).
The Fux' book is worthwhile since it is a historic text and it is referred to in so many contexts. If you decide to go for it, you should probably also look into this document which has useful hints for completing Fux' exercises:
There's one word of warning. That's that Fux' "gradus" does not teach you how to write counterpoint. It's mostly a series of excercises in avoiding things that do not sound well, not exercises that teach you to write what does sound well.
It's certainly useful, since it does helps to learn to appreciate all the different intervals, and it does help you to actively try and make melodies indpendent. It also helps you teach the basics of how to alternate consonances and dissonances and many of that theory still holds today, at least for tonal music. But it does not teach you about making effective themes, and it also does not teach you how about the harmonic implications of the counterpoint you write. (And often, you'll want to do that as well)
From around the site
from the tag wiki:
Also have a look at the search results:http://music.stackexchange.com/search?page=1&tab=relevance&q=counterpoint
the question What's "species counterpoint"? Are there any other types of counterpoint? is also relevant.
My take on Counterpoint
To my mind, Counterpoint is simply when you have multiple lines of music against each other at the same time. In the case of simply 2 lines of music played at the same time. A note on line 1 could be called a 'point', and a note played at the same time on line 2 could be considered the 'counterpoint'.
A simple case of all whole notes on both lines would be called Species 1 counterpoint, and a more complex example of whole notes on one line against half on the other would be called species 2.
For a great introduction to the topic of counterpoint I'd suggest:
The Study of Counterpoint by Johann Joseph Fux,
A series of posts on my blog about the book, and topic of counterpoint: Counterpoint on SheepishMusic
Counterpoint (from Lat. contrapunctus, from contra punctum: ‘against note’; Fr. contrepoint; Ger. Kontrapunkt; It. contrappunto).
A term, first used in the 14th century, to describe the combination of simultaneously sounding musical lines according to a system of rules. It has also been used to designate a voice or even an entire composition (e.g. Vincenzo Galilei's Contrapunti a due voci, 1584, or the contrapuncti of J.S. Bach's Art of Fugue) devised according to the principles of counterpoint.
Source: Klaus-Jürgen Sachs and Carl Dahlhaus. "Counterpoint." Grove Music Online. Oxford Music Online. Oxford University Press. Web. 17 Jan. 2014. http://www.oxfordmusiconline.com/subscriber/article/grove/music/06690