Often, but not always, these are allied with or part of a breed club. For instance, the Viszla Club of Northern California sponsors a Viszla rescue. The second type of rescue group takes in promising dogs from any breed or mix of breeds from a given shelter. Some of these groups do have purebred dogs. But these groups do not turn away a dog because of its indeterminate lineage.
Owners give up or abandon their dogs for a variety of reasons.
Sadly, many people still regard dogs as disposable, and will get rid of them if they are moving, or their schedule changes, or they fear the dog's reaction to a new baby, or they simply don't think the dog is cute anymore. Sometimes, the owners never take the dog to obedience training, then get angry at the dog for bad behavior. Shelters and rescue people will often work with these dogs before they are adopted to get them on the right track. Dogs that are truly dangerous or have major irremediable behavior problems are not put up for adoption. Dogs with disabilities or chronic minor health problems are sometimes available for adoption - it depends on the shelter or rescue.