Impromptu can be a noun, an adverb, or an adjective. It describes something as not being organized or planned in advance.
The plural form of the noun is impromptus.
Improvised is an adjective and is a synonym of impromptu. It carries the additional definition of being created with what’s around, or being makeshift.
In music, improvisation or an improvised piece is distinctly different than an impromptu piece. Improvisation is creating the music without sheet music or a plan, just letting it come out. This is common in jazz. Impromptu is more like recreating another piece without seeing the sheet music. It can also be called ‘playing by ear’. A person can hear a song and then recreate it. This is not ‘letting it come out’. A person can try many times to find the right melody and it is still impromptu.
Examples
In February, after several hundred people attended a protest against the construction of new hotels in the district, the soon-to-be hotel was occupied by activists, who turned it into a impromptu office to help those on the verge of being evicted from their homes. [The Guardian]
But on Sunday afternoon, his playing was truly convincing only in the repertoire he’s most associated with: a Liszt Hungarian Rhapsody and the most Romantic of the four Schubert impromptus on the program. [Boston Classical Review]
As a lieutenant in Afghanistan’s Helmand province in 2010, one of his squads was struck by an improvised explosive device. [The Denver Post]