The terms animal shelter, animal rescue, animal foster, and animal pound have been confusing to the average person since the beginning of time. The lines differentiating these groups blur with the laws and practices that each of these entities operate by.
Following you will find the short explanation of each type of entity, to the best of modern day definition.
Animals shelters are organizations that have physical structures to house stray and otherwise homeless pets. Animal shelters have a specific set of state laws that they must adhere to in order to "shelter", care for, and find homes for such great numbers of animals on an annual basis. Laws vary from state to state as do policies practiced from shelter to shelter. Animal shelters do not have animal control officers, although open door shelters commonly contract with city animal control offices to board their strays. Only rarely will shelters have an in-house cruelty investigator.
Keep in mind the subgroups of animal shelters. One being an open door shelter and the other a low or no kill facility. While the state regulations dictating both are identical, their policies on day to day operations vary greatly, with the obvious being their euthanasia policies.
An open door shelter takes in animals regardless of breed, health, or overall condition, and will usually keep the animals as long as the facility has space to hold them. Open door shelters will usually accept any animal, at any time, but do euthanize to create more space in the shelter when necessary. In public shelters euthanasia is common when space is needed.
A low or no kill organization does not euthanize to make space, but on the downside, is restricted by the number of animals they can take in when facilities are full and fosters are not available. They are not necessarily any pickier about the animals they accept, but are limited simply by numbers, according to their capacity. On rare occasions euthanasia may become necessary in cases of severe injury or incurable illnesses, otherwise animals at a low or no kill shelter are not euthanized, but treated for their afflictions.
Animal pounds are simply holding facilities for animals that are surrendered or picked up stray by an animal control officer (ACO). Pounds are required by law to humanely house stray animals until the state's stray hold time limit is up. The average in most states is five business days and it's during that period that pet owners may pick up their animals. Animals not picked up by their owners are promptly euthanized. Pounds are not required by any law or policy to re-home unclaimed animals, and few do.
Animal rescues, may or may not be actual physical facilities, but rather a network of pet fosters. Rescues take in owner surrenders and will pull, or take custody of animals in high kill shelters. Rescues take extra steps to ensure the welfare of an animal, beyond state laws and regulations, such as rehabilitation and long term care. Many, but not all rescues, are breed specific.
Pet fosters are individuals or families who care for and provide temporary homes for otherwise homeless pets. They do so through a rescue or shelter, until pets can be rehabilitated and readied for adoption. Fostering a pet can require a time frame of anywhere from a couple of hours up to several months, and in some cases, years. Fostered pets are normally more socialized than the average shelter pet and therefore more highly adoptable.