The term for high, florid, vocal singing involving scales, runs and ornaments. Sometimes these passages were written down, but often were extemporised by the performer. In this example from MozartÍs opera 'The Magic Flute', listen to the florid melismatic phrases which in places are almost an octave above the range of a normal female voice.

In vocal works the music illustrates the word or phrase, e.g. the words 'running down' might be illustrated by a descending scale. In this example from Handel's Messiah the words 'I will shake' are illustrated by a shaking in the notes.
Several notes sung to one syllable or vowel sound. In this example the melisma is on the 'joi' syllable of rejoice. Note that this can also be an example of coloratura and word painting. Sometimes all three of the above concepts can be applied to the same piece of music.
A technique used in vocal music where the singer is required to use the voice in an expressive manner halfway between singing and speaking. Many of the best examples are by Schoenberg from the start of the twentieth century. Listen to this example from his work 'Pierrot Lunaire'.
The English composer Sir William Walton also used this style in his work 'Facade'. Listen to a short example from that work which is in a much lighter style.