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Napoleon is said to have called Piazza San Marco "the finest drawing room in all of Europe" -- see Features below for the full story behind that line. Also known as St. Mark's Square, in honor of Venice's patron saint, it's one of the few large Venetian squares given over entirely to pedestrians, and technically the only one in the city actually called a "piazza" (every other Venetian square is a campo).
Many of Venice's most important buildings still stand on or near the square, including the Doge's Palace, St. Mark's Basilica and its clock tower, the former offices of the Procurators, and the Museum of Archeology. 
The square took shape in the 11th century, when a canal running through the area was filled in, and reached its present paved form during 16th- and 17th-century renovations, when geometric-patterned stone replaced the original brick -- a pattern that also helped mark out where traders' stalls once stood. The pavement was refurbished again in the late 19th century.
The smaller Piazzetta San Marco, between the square and the lagoon, was originally laid out in the 9th century and once served as a site for public executions; today a column topped by the lion of St. Mark watches over the crowds there instead.
Piazza San Marco began in the 9th century as a modest green space crossed by small canals, alongside vegetable gardens near the original St. Mark's Basilica. It didn't take on its current scale until the late 12th century, when Doge Sebastiano Ziani had one of the canals filled in and purchased additional land from a nearby convent, transforming the area into the large open square recognizable today.
For centuries, the piazza functioned as the ceremonial heart of the Venetian Republic: doges were formally installed here, naval commanders received their insignia before departing on campaigns and were celebrated with parades on their return, and the square hosted markets, festivals, and public gatherings that defined civic life in Venice.
Napoleon is popularly credited with calling Piazza San Marco "the finest drawing room in Europe," a line (real attribution unproven) that has stuck as shorthand for the square's grandeur -- an open, arcaded space ringed by some of Venice's most significant buildings rather than ordinary streets and shopfronts.
St. Mark's Basilica anchors one end of the piazza, its Byzantine domes and gilded mosaics facing directly onto the square, with the Campanile bell tower rising alongside it. Long arcaded buildings -- the Procuratie Vecchie and Procuratie Nuove -- line the north and south sides, historically housing Venice's civic administrators and now home to shops, cafes, and museums.
The piazza is Venice's lowest point, and it's the first place in the city to flood during acqua alta, the seasonal high tides that put parts of the square underwater; raised walkways are a common sight here during those periods. Thousands of pigeons share the square with visitors year-round, drawn by vendors selling seed to feed them -- fending them off (or feeding them) is practically a rite of passage for anyone crossing the piazza.
Historic cafes with outdoor seating and live orchestras have lined the square's arcades for centuries, giving the piazza its enduring reputation as Venice's social gathering place. Caffè Florian, which has operated continuously since 1720 and is considered the oldest coffee house in Italy still in business, once counted Lord Byron and Henry James among its regulars; its longtime rival Caffè Quadri, across the square, was reportedly favored by occupying Austrian troops for Turkish coffee in the early 19th century. Some cafes still charge a cover fee for the string-quartet performances that accompany an evening coffee at their outdoor tables.
St. Mark's Square is at its best in the early morning, before the crowds thicken, and takes on an almost eerie, misty quality in the fall and winter months. Some of Venice's best hotels, shops, and restaurants sit around the square, a gondola stop is a short walk away, and it's a common meeting point for tour groups and guided walks.
If a visit coincides with acqua alta, the square -- Venice's lowest point -- may flood; when it does, wooden walkways are laid down so visitors can still cross on foot.