Used to define a suburb of a big city that shares borders with it and, thus, acts more like a neighborhood of the city rather than it's own community.
Inner suburbs tend to have more nightlife, independent stores, and overall fun things to do than most normal suburbs, though (obviously) usually still have much less than the actual city. They also tend to have at least one decent mode of transportation into the city itself.
Yonkers, NY, Evanston, IL, Santa Monica, CA, and Cambridge, MA are all good examples of Inner Suburbs