Composers use forms in music. E.g. a fugue has strict rules. It has to have an exposition (the main theme) and further well defined developments of it. Not all forms are so strict, e.g. the forms for dance music are freer.
For improvisation the ternary form (song form) is suitable.
The simplest form has one sentence (A), consisting of 2 parts (a1 and
a2) like a question and answer. E.g. the Dutch song 'Little Ann sat in the
a1: Klein Anneke zat in de maneschijn, de maneschijn, de maneschijn,
a2: klein Anneke zat in de maneschijn, de maneschijn.
Hear the song.
The next song of Brahms has 2 sentences (AB) (a1 a2 b1 b2):
a1: Guten Abend, gut' Nacht, mit Rosen bedacht,
a2: mit Näglein besteckt, schlupf unter die Deck'.
b1: Morgen früh, wenn Gott will, wirst du wieder geweckt,
b2: morgen früh, wenn Gott will, wirst du wieder geweckt.
Hear the song.
Most common is the ternary form in 3 sentences, e.g. (ABA CB CB) or (ABA CDC ABA). A 'rondo' (a folk round dance) has a repeating procedure of two parts (refrain - verse) with a form of three sentences, usually as (ABACADAEA) or (ABACABA). This last one is a mixed ternary form with 3 sentences. You may read it as: (A-B-A C A-B-A). Mostly a ternary form has a definite closing, called coda.
An improvisation can't be as perfect as a composition, but some
systematic will give it an improvement. The use of an intro (a kind of
exposition but less strictly) will give it a finishing touch. And don't
forget to have some closings prepared for the same reason. The
last impression is taken home.
Hear 3 examples.
An intro may also be used to put one song through to the next. Moreover, it is easy. You get the time to decide what the next number will be, in which key and tempo and to memorize how the melody and the chord scheme were. An intro may also be used to mask a little meditation for inspiration.
If you accompany a singer, you may use the intro to help him with the first notes. A good companion is valuable for a singer. Everyone who has sung knows how terrible a bad start is.
How do you make an intro? The overture of an opera contains elements of what is going to come like the index of a book. This principle may be used for improvisation too. But it all depends on the situation, e.g.: