I know there are lots of animial shelters in the US and other places in the world. Are they not all run by rescue organizations? If not, what is the difference between a shelter run by a rescue organization and one run by a government entity?
Some animal shelters are run by local government. These shelters are supported by tax dollars (though some also have non-profit arms to raise additional funds if the government funding is not enough for their operating costs).
Generally, government shelters are operated as open access (an animal can be dropped off regardless of the number of animals currently at the shelter), since they traditionally operate as community lost and found stations for lost pets.
Private shelters (run by rescue organizations) can be open access or limited, depending on the organization. They may function as a community lost and found station if there is no government shelter, but in my experience (and in some reading I did when I found Agent Orange a few weeks ago) if there is a local government shelter they generally prefer you to take found animals there to have a central location for owners.
Other community shelter functions (educational seminars, low cost spay/neuter/vaccination events, etc) are offered by both government and private shelters, depending on their resources. Both types of shelters need volunteers to do basic functions like clean cages and socialize the animals.
(For reference, I've volunteered at my local government shelter. I've visited my local private shelter and the private shelter associated with the rescue group I work with (most of their cats are home fostered, but they have a small facility in central VA)).