Stravinsky’s Suite Italienne is a reworking of his ballet Pulcinella. This specific version was written for violin and piano in 1933, where the complete ballet is from 1920. Stravinsky wrote the ballet for Diaghilev’s Ballet Russes in Paris, and it was meant to be based on fragments of the Italian Baroque composer Pergolesi. That is was actually not written by the man doesn’t matter for the Italian atmosphere of the ballet, and by extension this little suite.
By Stravinsky’s time the term suite meant something else than in the Baroque age, when it was a series of dances, usually six. Here we have a late romantic suite. These were created like shorter, more accessible arrangements of a longer work. The original can be an opera, a ballet or maybe incidental music. These longer works were not for everyone, so they were shortened. Kind of a best-of I suppose.