The hansom cabriolet was introduced to the city in the 1830s, and stayed on its street for a century. Although most were replaced by motor vehicles after the First World War, the last horse cab licence was only surrendered in 1947.
The driver's position is a little surprising, on top and to the rear of the vehicle (but perhaps less so than the original design, which had him sat on the roof directly above his passengers). Below and before him, two passengers could be comfortably seated in an enclosed cab. A hatch in the roof let them speak to him and pass up money for the fare. As for the amount to be paid, that did not have to be determined by taximeter until 1907 although such meters began to appear in the nineteenth century. The first models were clockwork, very different to the entirely electronic versions used today. It is the use of such a meter which makes a cab a taxi.
Although this Pathe newsreel depicts a wedding in Sydney, Australia, it does have very good footage of a hansom cab:
Further reading: here's a great description of a cab stand in 1860.