In the United States, there are two sorts of foreclosure in most common law states. Using a “deed in lieu of foreclosure,” the bank claims the title
and possession of the property back in full satisfaction of a debt,
usually on contract. In the proceeding simply known as foreclosure (or,
perhaps, distinguished as “judicial foreclosure”), the property is
exposed to auction by the county sheriff
or some other officer of the court. Many states require this latter
sort of proceeding in some or all cases of foreclosure, in order to
protect any equity
the debtor may have in the property, in case the value of the debt
being foreclosed on is substantially less than the market value of the
immovable property (this also discourages strategic foreclosure). In
this foreclosure, the sheriff then issues a deed to the winning bidder
at auction. Banks and other institutional lenders typically bid in the
amount of the owed debt at the sale, and if no other buyers step
forward the lender receives title to the immovable property in return.

Statutory foreclosure is foreclosure by performance of a power of sale
clause in the mortgage without need for court action, since the
foreclosure must be done in accordance with the statutory provisions
governing such sales.

Strict foreclosure
refers to the procedure pursuant to which the court ascertains the
amount due under the mortgage; orders its payment within a certain
limited time; and prescribes that in default
of such payment a debtor will permanently lose his or her equity of
redemption, the right to recover the property upon payment of the debt,
interest, and costs. The title of the property is conveyed absolutely to the creditor, on default in payment, without any sale of the property.

Resources:

HUD – How to Avoid Foreclosure

Stop Foreclosure – Sell Your House

Wikipedia – Foreclosure