As with all hummingbirds, Anna's have a very rapid wing beat which produces a humming sound. They can fly very fast forward but can also fly sideways, up, down and backwards, similar to a helicopter. In fact, they feed almost exclusively while hovering. Natural food consists of nectar from tubular shaped flowers supplemented by small insects and spiders. We suburbanites often attract hummingbirds with sugar-water syrup; it is important to keep the syrup fresh and in the correct concentration, about 1 part sugar to 4 or 5 parts water.
The Anna's song consists of a repeating series of short tunes, usually a perched male announcing himself. When defending his territory, a male will blast out a rapid series of warning chirps and chase the intruder quite some distance. During nesting season, one male might be busy chasing an intruder (male or female) while a third bird (usually a female) will come in for a quick meal. (See http://animaldiversity.ummz.umich.edu/accounts/calypte/c._anna$narrative.html for some samples)
Anna's hummingbirds are quite bold and will often hover a few feet away (about 1 meter) from humans within their territory. I was once changing feeders when a bird can to feed; I stood motionless while the bird drank from the feeder I was holding. They tend to ignore all other types of birds, as they do not compete for food. They seem to be particularly attuned to the sound of other hummingbirds, and upon seeing or hearing another hummingbird, will attempt to chase it off.