Before the bus there was the stagecoach and the first services we could call "buses" were of a similar vein. The difference between a stagecoach and a bus is generally that the latter does not require booking in advance and people can alight throughout the journey, though in the earliest days the technology was the same. The first such service began in Manchester in 1824 thanks to John Greenwood. Omnibus services (as they became known) spread to the other major cities and beyond including the first service in London operated by George Shillibeer in 1829  between Marylebone and the Bank of England.
These buses were horse drawn although there were experiments with steam traction as early as 1830. Government legislation virtually banned the development of motor buses until 1896. This ensured that until the turn of the twentieth century the horse bus reigned supreme. By 1900 there were over three and half thousand horse buses in London alone.
The early days of omnibuses were somewhat wild. Fierce competition led to buses racing each other to compete for passengers. Licences were introduced for drivers and conductors in 1838 in London following complaints by passengers . Initially horse buses were single decker though roof seats began to be introduced in the 1840s as people were climbing onto the roofs of buses anyway so operators thought they should try and get some money out of it!
Experiments with motorised buses began in 1897 with experimental services employing petrol engine buses. Horse buses remained in service for a number of years but their days were numbered. Services ended in London in 1914 as the horses were unfortunately needed for a sadder purpose: the First World War. Outside of the capital horse buses remained in some areas until the early 1930s.
|London Omnibus (National Archives )|
|London General Omnibus Company 'garden seat' type horse bus, circa 1881 preserved at LTM Acton|
|Shillibeer bus preserved at London Transport Museum|
 Horse buses in London http://content.tfl.gov.uk/research-guide-no-14-horse-buses-in-london.pdf
 Philip Bagwell & Peter Lyth, Transport in Britain 1750-2000 (Hambledon, 2002) p. 106