Horse-drawn omnibuses ran along a fixed route in a city. They were meant to seat fifteen passengers, although they were often over-crowded. This photo is of a horse-drawn omnibus used today as a tourist attraction in Antwerp, but it gives you a pretty good idea of what the experience must have been like.
Eventually they put steel rails in the streets and omnibuses were replaced by horsecars. Not to be confused with cable cars (which were sometimes steam driven).
How did a passenger on an omnibus let the driver know that his or her stop was coming up? Any guesses?