The piece that Mary plays at the party at Lucas Lodge in episode 1, section 3, is popularly called the "Harmonious Blacksmith", from the Suite #5 by George Frideric Handel.
The piece that Caroline Bingley plays at Netherfield in episode 1, section 6, is popularly called the "Gypsy Rondo", from the Piano Trio #39 by Joseph Haydn
The first piece that Mary plays and sings (badly) at the Netherfield party, in episode 2, section 5, is popularly known as "Handel's Largo" and is the opening aria of the opera "Serses". It is called 'Ombra mai fu'. She then goes on to sing "My mother bids me bind my hair", an art song by Haydn (and actually much less ambitious given it was written for the purpose of performance in the home).
The piece that Louisa (Bingley) Hurst then plays (perhaps to show Mary up) is the "Rondo alla Turca", the 3rd movement of the Piano Sonata #11 by Mozart.
The piece that Lizzy plays for Colonel Fitzwilliam and Mr. Darcy at Rosings, in episode 3, section 4, is the 1st movement of the same Mozart sonata that contains the "Rondo alla Turca".
The song that Lizzy sings and plays at Pemberley, in episode 5, section 1, is 'Voi che sapete' (translated here as 'Say, ye who borrow') from Mozart's "The Marriage of Figaro". The piece that Georgiana then plays, in the scene when Lizzy and Darcy's eyes meet, is Beethoven's "Andante favori".