The Oxford Concise Dictionary of Music by Michael Kennedy (1996) defines Improvisation as a performance according to the inventive whim of the moment, i.e. without a written or printed score, and not from memory. It has been an important element in music throughout the centuries.
Improvisation is something quite interesting and fun to do. It is an amazing ability to be able to sit down at a piano, or whatever instrument, and just play to your heart's content.
However, this ability requires an understanding of music theory, so I suggest that you (the reader) go to the beginner section and read the section on introductory music theory. If you feel that you are comfortable with music theory, then please continue.
In classical music we have what is called a cadenza. A cadenza is an improvization, showing off the virtuosity of the soloist (whether it be a singer, pianist, violinist etc...), which is usually inserted into the final cadence of any section of a vocal aria or solo instrumental movement.
Basic improvisation relies largely on the accompaning chords. Generally, to basically improvise is to take the notes of the chord being played and mix them up, add a rhythm, and turn them into a melody. Here's an example. Let's say we have a four bar chord progression. C|F|G|C In the first bar you can take the three notes of C, and mix them up. Let's say in this example you play one note per beat, with four beats per bar. It means that you could randomly use C,E or G in an order, which you would make up as you went. In the second bar you could mix up the notes F,A and C in any order, and in third bar using the notes G,B and D. Here is an improvisation. Try playing this on your instrument: C G E C|C F F A|G B D B|C E C C You can mix up the notes in any order to make an improvisation.