The origin of the term suburb is from the Latin suburbium, which is a compound word combining "sub" (below) and "urbs" (city). Rome was built on the famous seven hills, and suburbium meant the area outside the city walls.

The furthest edges of the suburbs became a place where the wealthy built country homes. Catullus 44 says: "fui libenter in tua suburbana uilla, malamque pectore expuli tussim" which can be translated as "I was glad to be in your suburban villa and to clear my chest of the troublesome cough".

The term has since been applied to other cities.

There is more to read at The Literary Allusions to the Suburbium of Rome and their Social Implications

Which kind? Various kinds of suburbs exist in the developed world and I am not sure which kind you talk about.

There is the tract housing type most typical in North America which is largely about building on formerly empty land, typically a cooperation between urban planning and real estate business building a lot of them in a row. For this first type look up things like electric railways, Henry Ford cars, and interstate highways on Wiki.

Then you have the former villages getting integrated into the city type of suburb, this exists in North America as well (see Greenwhich Village) but basically most major European cities are made like this.

These villages have existed for a really long while, as they provided food for the cities. The idea of cities holding market days and neighboring villagers bring in food those days is essentially Medieval. They later on got incorporated into the cities but then just further and further villages serve as suburbia. To tell when did these villages start acting as a suburbia, as sleeping places for people working in the cities, so basically when commuting got popular, agan, look up various kinds of modern transportation, cars, light rail, and the most interesting aspect - horse pulled omnibus service, which made living somewhere else than you work possible. "Bus" is just a short form of omnibus, and omnibus means "for all".

The suburb, broadly speaking as the suburbs we know today, were first built in response to overcrowding in cities. With the rise of modern commerce and urbanization in the 18th century, cities began to be portrayed as dirty, hotbeds of disease and revolution and to some extent this was true. Take "Pope Day" in colonial Boston for example:

Tuesday last being the Anniversary of the Commemoration of the happy Deliverance of the English Nation from the Popish Plot... It has long been the Custom in this Town on the Fifth of November for Numbers of Persons to exhibit on Stages some Pageantry, denoting their Abhorrence of POPERY and the horrid Plot which was to have been executed on that Day... the Servants and Negroes would disguise themselves, and being armed with clubs would engage each other with great Violence, whereby many came off badly wounded; in short they carried it to such Lengths that two Parties were created in the Town, under the Apellation of North-End and South-End

In essence, the people of Boston would get so drunk that they would literally beat the shit out of each other based on where they lived geographically. This tradition carried on well through the Colonial Era and no one dared intervene. 18th century London was equally rowdy and Paris was, obviously, a place given to riot given the French Revolution.

This same fear of revolution and disease prompted several other early urban renewal projects, such as Haussmann's Renovation of Paris and Stewart's Garden City. Stewart's plan was the first attempt to create something like a modern suburb -- his towns were zoned by residential, commercial and agricultural spaces all centered around a rail link -- and the premise caught on. In general these early suburbs connected by rail were generally subsumed by the larger city once they grew to sufficient size until the Town of Brookline refused annexation in the mid-19th century. This was the first true suburb.

Suburbs built in the mid-19th century to the end of the Second World War continued to be very walkable places. After the passage of various reforms in 1945, developers began to build tract housing in order to alleviate what was perceived as overcrowding in American cities. Many Americans found a quarter-acre plot with a small ranch house to be infinitely preferable to living in a two-bedroom tenement in their major city and in the subsequent years hundreds of millions of Americans would abandon the cities they grew up with in order to live in what were generally homogenous, class and race based suburbs only accessible by car.

I remember learning about the history of suburbs in the U.S at least, really started when the rich wanted to move out of cities and separate themselves from everyone else in the country. They were the only ones that could afford the type of transportation to and from the cities and newly created suburbs which were only a few miles outside the actual city compared to suburbs these days which are further out into the country. Suburbs began to become more intriguing and popular to everyone else once trains and cars were finally produced which allowed more people to move out of cities and settle into suburban type areas.