I don't think putting down a PET dog that wasn't used for breeding because of age/illness would be considered culling. Some breeders do cull pups and dogs they use strictly for breeding tho, but not ALL are killed. It's easy to "cull" a dog from a breeding program by getting them sterilized and adopted out to a pet home. The example that comes to mind is white boxers. People who irresponsibly (in my opinion) breed double merle/dapple dogs end up with lethal white pups as a byproduct, and those usually have to be put down because they are generally too miss-formed to have any sort of normal life.
Culling is really the removal of an animal from a breeding program, just like Rhasputin said. They do "culls" in wildlife management too using permit hunts with wolves, boar, deer, etc. It's all about control of population and health.
In mice, culling generally comes in 2 forms, which is either to end the mouse's life, or to adopt them out as pets. Culling the runts from litters leads to healthier litters overall, and most fanciers could never keep all the mice they produce. Also, there aren't enough trustworthy homes to give every mouse ever born to someone to care for. There are other ways and reasons people cull, but those are the most common. It is each breeder's job to come up with their own ideas of culling that will work with them and their mice, and not everyone is the same. Some people cull very heavily and some never do at all.