Giardia are single-celled parasitic organisms that infect many types of animals around the world. Humans can become infected as well, but it is rare for Giardia from dogs and cats to transmit to people, and the same is true for transmission between dogs and kitties.
Organisms are passed into the environment in feces and once outside the body, they turn into hardy tiny cysts that can survive for months. Once they are ingested by a new host (for example from contaminated water, or a dog licking her feet after walking in the grass) the shell dissolves and every cyst releases two infectious organisms. Once free from the cyst, Giardia "swim" around inside the host's intestines until they find a good spot to feed and become attached to that area. Once there, they can move around in different parts of the intestines looking for different nutrients.