A very small urban park (small enough to fit into someone's vest pocket?), usually found on a leftover scrap of land accessible to a street.
Towards the end of the 19th century, the so-called "safety valve" of American westward expansion was outstripped by the twin forces of industrialization and immigration. By 1900, at least 40% of all Americans lived in cities. This was a period of unbelievable poverty and overcrowding in cities, leading to filth, disease, crime, and misery in general. Many reformers saw pointed to the urban environment itself as a cause: Harsh concrete and stone buildings caused stress. The open sewers ironically called "streets" were a direct hazard to public health. The City Beautiful Movement, which was at its peak from about 1890 to 1910, drove countless public works and public health projects. As a reflection of the "back to the winderness" philosophy of The Romantic Movement, the introduction of green recreational spaces was seen as the cure for a multitude of urban malaises. And so, well-known landscape architects such as Frederick Law Olmsted were hired to design vast expanses of manicured lawns, walkways, and benches, peppered with fountains and spectacular Beaux Arts edifices.
But the value systems that drove the creation of large parks came from a time before the automobile made its impact on American culture. Citizens were expected to walk or ride the streetcar to their green spaces. As the 20th Century wore on, and more people got around by car, it was possible to drive out of the city to see the actual countryside. Walking several miles to visit an imitation seemed second-class.
After World War II, the demise of the streetcar and the increased demand for quality housing caused people to move out of the cities altogether. Towards the end of the 1950s, urban planners could see that large parks were not serving the open space needs of the remaining city dwellers. At the same time, the exodus caused the familiar problems of American urban life to resurface: poverty, crime, drugs, poor schools. However, overcrowding was replaced with decay as more and more people moved out of the central city. The Open Space Movement, which began around 1960, saw this arising from the same need for green spaces that created the big parks. This time, however, they would have to be nearby, visible from a front door or only a few minutes' walk away. In the new planned communities which were cropping up around the country, land was available to build greenways and greenbelts, and to design land developments with green, open spaces built in.
This was not the case in the inner cities: Most of the land was built upon, or paved over. However, every city had its nooks and crannies: The land left over from building a subway station. A remnant undeveloped lot. A decrepit building, or the sites of a recently-demolished building. A scrap of land that was shaped too oddly to be worth the cost of improving. Planners saw an opportunity to create green spaces and recreational facilities in many small places instead of a few large ones. It was thought that small, attractive areas would make a city's old neighborhoods attractive again. And so, vacant lots were converted into playgrounds for neighborhood children or minature gardens with fountains and places to sit.
The term "vest-pocket park" is usually applied to any recreational or green space smaller than 3 acres (or about 1 hectare). I'm not comfortable with this measurement, since most of the ones I'm familiar with are much smaller. I also see a variation on the basic concept: A large, mostly-wooded urban park is frequently the site of a project which revitalize a small section for a special purpose (cf. Mother's Garden in Baltimore's Clifton Park, or Strawberry Fields in New York City's Central Park). Might these not be considered vest pocket parks as well?
The benefit from these little parks has been mixed. Continued urban flight in the 1970s and 1980s dramatically reduced city tax bases, forcing them to make difficult budgetary choices. Frequently, it was decided to cut back on the maintenance of parks, and the remaining money was concentrated on the more visible large parks. Many small spaces began to become overgrown with weeds, piled high with trash, places for drug dealers to meet, places to stash the growing number of homeless people and forget about them. Places that were supposed to draw people to a community became a reason to stay away. Through the succeeding decades, the fate of the vest-pocket park depended on the vagaries of mayors deciding to temporarily "clean up" a neighborhood for its publicity value, or residents of particularly strong communities who took matters into their own hands.
You will still find vest-pocket parks scattered in cities throughout the industrialized world. Mixed in with the big parks, they constitute the green spaces that make urban life not quite so oppressive.