That's a good question...
...because they all are so different! Basically, a residency is a place that offers you space and time to create and think. It can be a place of production as well as reflection. They can be located in major cities or small towns, or somewhere in between. Some offer funding to residents: grants, stipends and even included meals. They can range in time from a couple days to a year or more.
Alicia made an infograph illustrating what an artist residency can be: