Once a domain name has passed its expiry date, a number of things happen. Firstly, the domain name is typically put on hold - that is, its name server information is deleted or modified so that the domain name no longer points to the website it is associated with.
Most registrars have a "grace period" (sometimes detailed explicitly on their site or by email, oftentimes applied without comment) after domain names have expired.
During that grace period, the original owner of the domain name can pay to renew their domain name (and hence remove it from "on hold" status and reactivate it). Some registrars may impose an additional administrative "penalty fee" to renew domain names during their grace period.
At the end of the grace period, the existing owner can no longer renew their domain name and has lost all control over it. What happens next depends on the registrar...
Some registrars will delete the name immediately following the end of the grace period. Some will hold it for a certain additional time-period before releasing it. And some will change the ownership information on the domain name so that it becomes registered to the "Unpaid Names Department" or similar, and continue to hold the name for an extended period of time before it is finally deleted..
Once a domain name has reached the point where it is about to return to the domain market i.e. it will once more become available for registration. If the domain name is considered valuable, there may be many interested parties lining up to try and grab it i.e. to attempt to secure it as it is deleted.