When a domain reaches the end of it's registration, the domains usually go up for auction.
If nobody buys it the domain will be dropped and available to be registered by anybody.
Not all expired domains go for auction. If you are looking for an expired domain name just go to some registars homepage and simply try to register it again. Registars registration page will tell you if it's available or not.
People try to backorder expired domains and most wanted ones will go to auction. Depending who snaps it first.