A sonata is usually a piece in several movements (sections) for a solo instrument (most commonly piano) or solo instrument with piano accompaniment (most commonly violin and cello, but can be any instrument). Sonatas are nearly always for one or two instruments.
An overture started off as an introductory piece of music played before an opera. Operatic overtures sometimes (but not always) contain tunes which will be heard later in the course of the opera. Most (but not all) operas have an overture (sometimes called a 'prelude'). Examples:
Beethoven - 'Leonora' and 'Fidelio' overtures
Mozart - overtures 'The Marriage of Figaro', 'The Magic Flute', 'Don Giovanni', 'Il seraglio'.
Smetana - overture 'The Bartered Bride'
Wagner - overtures/preludes 'Lohengrin', 'Tannhäuser', 'The Flying Dutchman'
Weber - overtures 'Oberon', 'Der Freischütz', 'Euryanthe'
There is another kind of overture - the 'concert overture' - which is a stand-alone piece usually of a programmatic nature (ie it relates some kind of narrative, story or theme). Examples:
Beethoven - 'Namensfeier', 'The Consecration of the House'
Dvořák - overtures 'Carnival', 'Othello'
Mendelssohn - 'The Hebrides (Fingal's Cave)', 'Buy Blas'
Tchaikovsky - '1812'
Overtures are written for and played by an orchestra.
A 'suite' is a selection of 3-5 (sometimes more) sections of music linked by a common programmatic (usually) theme. Composers sometimes extract suites from operas and ballets so some of the music from a much longer piece can be included in a concert (eg Tchaikovsky's 'Nutcracker' suite - a 20-minute suite extracted from a ballet lasting nearly an hour and a half or Grieg's 'Peer Gynt' suites - extracted from a two-hour play with incidental music).
Sometimes suites can be stand-alone works too, but nearly always have a narrative theme linking all the movements.
Suites started-out as collections of stylised dances in the 18th century with an overture (remember that?) preceding them (eg the Orchestral Suites by JS Bach), but these soon evolved into something more colourful and programmatic.
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