The Science research contributed by a city is indicated by the light bulb symbols in the resource window of the city display. The science from a city is added to previous research and, if enough research has been done, a new civilization advance is achieved. The amount of science a city produces is mainly determined by the science rate of your civilization, the percentage of each city's trade that arrives as new knowledge. You can modify this rate by adjusting your science rate from the Game menu.
You can also increase science by building Library and University improvements, certain Wonders, and by turning some of your citizens into Scientists.