The nonjudicial foreclosure goes in California goes like this:

Borrowers approach bank for loan.  Bank has borrower sign promissory note. 

Bank also has borrower sign a Deed of Trust which puts the house (lot and improvments) up  as collateral against the loan value.

Bank has the Deed of Trust recorded in the Official Records Office of the County where the property is located.

Borrower stops paying.

Bank sends friendly reminder and overdue letters by mail to the borrower.

Bank later tells trustee to file a Notice of Default on the Official Records showing amount overdue.

Bank later tells trustee to file a Notice of Trustee Sale that sets a date, time and place (County courthouse steps usually) for auction.

Naturally, Borrower continues to NOT pay during this scenario.

Trustee hires auctioneer to handle auction on appointed date.  Auction day arrives and the auction occurs.  Bank is ALWAYS first bidder as that sets the initial bid price.

House is "under water" (amount due is higher than value of home in current, declining-value market conditions) and nobody else bids.  Bank owns property.

Trustee has a couple days to record Trustee Sale Deed into bank name.

Bank can evict Borrower or allow Borrower to continue to live there.  Bank chooses latter as they can't unload the house under current market conditions.  Better to have a (former) owner continue to live in the home and upkeep it than get cited $1k daily for neighborhood blight, zoning violations and lawsuit potential from accidents that might happen.

Someday, bank evicts former Borrower, either by mutual agreement or by court order and sheriff enforcement with notice posted on door and the like.  Long costly process if the former Borrower does not move willingly.

This varies for actual mortgages, property tax foreclosures, tax seizures, and homeowners association foreclosures.   Complicating factors (but not by much) might include promissory notes that are sold (and hence, should be re-assigned of public record by recording an Assignment of Deed of Trust), or rental units under lease, or commercial  foreclosures.  Each other scenariou offers nuances to the nonjudicial foreclosure process, as you can imagine. 

Also, the scenario above is very simplified and does not include auction delays, filing deadline issues, flawed notification issues and more.