House mice are the same species as field mice (Mus musculus). They are not native in the Americas; they originated in India. There are many native rodents, including native mice; however, native mice belong to a different taxonomic family than common house mice.
When land developers move in, field mice have to move out. And what better place to move to than into your nice warm house, with food to nibble and sweaters to shred and counters to poop on and comfy walls to build nests in? Many families of house mice have been living in human homes for generations and have different social traits. In Europe they display notable physical traits, so your house mouse could be the ancestor of a future distinct Californian house mouse.
When you find yourself overrun with unwanted mouse guests, there are many ways to encourage them to leave. First, keep your house extra clean — all food should be tightly stored. Then head over to Ace Hardware and pick up your accessories of choice. Live traps are popular in Davis; they allow you to capture the mouse without harming it, and then you can release the mouse back into the wild to become a field mouse once again. In extreme situations, poison or snap traps may be necessary. Glue traps are effective, but fairly cruel to the mouse — where poison and snap traps at least give the mouse a quick death, glue traps stick the poor creature to a piece of cardboard to slowly starve to death. Another option for traps available at Ace is the "Rat Zapper 2000". It is the most expensive trap but a good one. It kills both mice and rats (unlike spring traps) using the charge from a few AAs. You don't have to see your kill (it's hidden in a little tunnel) and you keep poison out of the food chain. They cost about $50.
If you really want a totally humane way to dispatch a mouse, halothane or slow CO2 inhalation (not dry ice!) are the generally accepted methods among the rodent enthusiast community.
House mice whom you invite to live in your home, for whom you visit Petco and buy fancy wheels and salt licks, would be more properly characterized as Friends of the Wiki. It is important to distinguish the two, especially since releasing a domesticated mouse into the wild ensures a very nasty death for the animal, and taking a wild animal out of their habitat isn't the best idea as rodent domestication takes several generations.
For more general information on our tiny fuzzy co-inhabitants, check out the mice entry.