The groundwork for a lot of the way we think about public space was laid by social ecology and environmental psychology thinkers of the mid- to late-20th century. If you're not already familiar with these names, have a look at some of the links here, and if you'd like, check out their original writings.
The proto-urbanist of the modern city is Lewis Mumford. He had a way with words--one of my favorite quotes of his is "forget the damned motor car and build the city for lovers and friends." His writings were the beginning of the critique of the engineered, technocratic city that would later become the populist defense of Greenwich Village against freeways and urban renewal made by Jane Jacobs. Her most important book, The Death and Life of Great American Cities, gets into the social life of neighborhoods and explores the kind of informal social interactions that make for resilient, safe, interesting places. New York's Project for Public Spaces considers Jacobs its intellectual godmother. Her defense of "eyes on the street" and informal play on stoops and in alleyways ran against the grain in the 60s, but is common wisdom in planning today.
Allan Jacobs, no relation to Jane, is a designer at emeritus faculty at UC Berkeley. His book Great Streets outlines the design characteristics of good public spaces and looks at how scale and form contribute to the social life of cities.
The person to have done the most to create primary research on how people actually use public space is William Whyte. His contribution was to systematize and quantify the successful design characteristics of public spaces (much like Corbusier did for architecture in creating the proportions of his Modulor Man) by actually filming people using public space and scientifically analyzing their behaviors. Whyte's work was tremendously influential and helped urban designers understand the human implications of the spaces they were creating.