It refers to the use of a major chord of the tonic at the end of a musical section that is either modal or in a minor key. This is achieved by raising the third of the expected minor triad by a semitone to create a major triad, as a form of resolution.
For example, instead of a cadence ending on an A minor chord containing the notes A, C, and E, a Picardy third ending would consist of an A major chord containing the notes A, C♯, and E. Note that the minor third between the A and C of the A minor chord has become a major third in the Picardy third chord.
Even in instrumental music, the picardy third retains its expressive quality: it is the “happy third.”…Since at least the beginning of the seventeenth century, it is no longer enough to describe it as a resolution to the more consonant triad; it is a resolution to the happier triad as well.…The picardy third is absolute music's happy ending. Furthermore, I hypothesize that in gaining this expressive property of happiness or contentment, the picardy third augmented its power as the perfect, most stable cadential chord, being both the most emotionally consonant chord, so to speak, as well as the most musically consonant.
What makes this a Picardy cadence is shown by the red natural sign. Instead of the expected B-flat (which would make the chord minor) the accidental gives us a B natural, making the chord major.
The origins of the term are obscure. An idea that was repeated as fact for some time, but turns out to have no provable basis, was that expounded by Jean-Jacques Rousseau in his Dictionnaire de Musique (1767): that this form of ending survived longest in church music, and due to the great number of cathedrals in the historical French province of Picardy. More plausible is the idea that the North of France, and Flanders, were influential in the development of contrapuntal music in the fifteenth century.
Robert Hall hypothesizes that, instead of deriving from the Picardy region of France, it comes from the Old French word "picart," meaning "pointed" or "sharp" in northern dialects, and thus refers to the musical sharp that transforms the minor third of the chord into a major third. In medieval music, such as that of Machaut, neither major nor minor thirds were considered stable intervals, and so cadences were typically on open fifths. Examples of the Picardy third can be found throughout the works of J.S. Bach and his contemporaries, as well as earlier composers such as Thoinot Arbeau and John Blow.
This practice began to decline in the late sixteenth century and by the Classical era had been more or less discarded, although examples can be found in works by Haydn and Mozart. However, the switch from minor to major was a device frequently used to considerable expressive effect by Schubert. In “Gute Nacht”, the strophic song that opens the song cycle Winterreise, Schubert changes from minor to major for the last verse:
(This passage occurs at 4:05 on the linked recording.) In his book on the Winterreise, singer Ian Bostridge speaks of the “quintessentially Schubertian effect in the final verse, as the key shifts magically from minor to major.”  In the Romantic era, those of Chopin's nocturnes that are in a minor key almost always end with a Picardy third. A notable structural employment of this device occurs with the finale of the Tchaikovsky Fifth Symphony, where the motto theme makes its first appearance in the major mode.
It is notable that in the first book of J. S. Bach's The Well-Tempered Clavier composed in 1722, only one of the twenty-four minor movements fails to end in a Picardy third, whereas in the second book, composed in 1744, fourteen end without it. (Manuscripts vary in some of these cases. This is the case with the single exception in the first book, the G♯-minor fugue, which, according to the present Bach Gesellschaft edition, is thought to have been originally composed in G minor, accounting for the unusual enharmonic spelling of the final G♯ major chord as G♯–C♮–D♯ instead of G♯–B♯–D♯.)[original research?]
According to James Bennighof: "Replacing an expected final minor chord with a major chord in this way is a centuries-old technique—the raised third of the chord, in this case G♯ rather than G natural,[verification needed] was first dubbed a 'Picardy third' ("tierce de Picarde") in print by Jean-Jacques Rousseau in 1797... to express [the idea that] hopefulness might seem unremarkable, or even clichéd."
- Beatles - "I'll Be Back", from the soundtrack album of the film A Hard Day's Night. Ian MacDonald speaks of the way “Lennon is harmonised by McCartney in shifting major and minor thirds, resolving on a Picardy third at the end of the first and second verses.” 
- Beethoven – Hammerklavier, slow movement
- Brahms – Piano Trio No. 1, scherzo
- Dvořák – New World Symphony, finale
- Joni Mitchell – "Tin Angel", from Clouds (1969); the Picardy third lands on the lyric "I found someone to love today". According to Monk, the Picardy third in this song "suggests Mitchell is internally aware of romantic love's inability to provide true happiness..."
- Bob Dylan – "Ain't Talkin'", the final song on Modern Times (2006), is played in E minor but ends (and ends the album) with a ringing E major chord
- Roberta Flack – "Killing Me Softly With His Song" ending and resolution. According to Flack: "My classical background made it possible for me to try a number of things with [the song's arrangement]. I changed parts of the chord structure and chose to end on a major chord. [The song] wasn't written that way."
- Bruce Benward and Marilyn Nadine Saker, Music in Theory and Practice: Volume II, eighth edition (Boston: McGraw-Hill, 2009), p. 74. ISBN 978-0-07-310188-0.
- Percy Scholes (ed.), The Oxford Companion to Music: Self-indexed and with a Pronouncing Glossary and Over 1,100 Portraits and Pictures, ninth edition, completely revised and reset and with many additions to text and illustrations (London and New York: Oxford University Press, 1955), pp. 1027–28.
- Peter Kivy, Osmin's Rage: Philosophical Reflections on Opera, Drama, and Text, with a New Final Chapter (Ithaca and London: Cornell University Press, 1999), p. 289. ISBN 978-0-8014-8589-3.
- Denise LaGiglia and Anna Belle O'Shea, The Liturgical Flutist: A Method Book and More (Chicago, IL: GIA Publications, 2005), p. 166. ISBN 978-1-57999-529-4.
- Robert A. Hall, Jr., "How Picard was the Picardy Third?", Current Musicology 19 (1975): pp. 78-80.
- Ian Bostridge, Schubert's Winter Journey (London: Faber and Faber, 2015): 7.
- James Bennighof, "The Words and Music of Joni Mitchell", Santa Barbara: Praeger, 2010.[page needed]
- Ian Macdonald, Revolution in the Head: The Beatles Records and the Sixties (London: Pimlico, 2005): 119.
- Robert S. Hatten, Musical Meaning in Beethoven: Markedness, Correlation, and Interpretation (Bloomington: Indiana University Press, 1994), p. 39. ISBN 0-253-32742-3. First paperback reprint edition 2004. ISBN 978-0-253-21711-0.
- Johannes Brahms, Complete Piano Trios ([full citation needed]: Dover Publications, 1926),[page needed]. ISBN 048625769X.
- Antonín Dvořák, Symphonies Nos. 8 and 9 (Dover Publications, 1984), pp. 257-258. ISBN 048624749X.
- Katherine Monk, "Joni: The Creative Odyssey of Joni Mitchell" (Vancouver: Greystone, 2012) p. 73. ISBN 9781553658375
- See Ain't Talkin in Songs list at dylanchords.info. The guitar part is played in Em with a capo on the 4th fret, so the song sounds in the key of G♯ minor.
- Toby Cresswell, 1001 Songs (Pahran, Austria: Hardie Grant Books, 2005), p. 388, ISBN 978-1-74066-458-5.
- Ruff, Lillian M. 1972. "Josquin Des Pres: Some Features of His Motets". The Consort: Annual Journal of the Dolmetsch Foundation 28:106–18.