After a domain expires, it will go into quarantine (you can not use it any more). This is a period when you can still renew it, but additional cost may apply. During this period domain can not be registered by someone else. As soon as quarantine period is over, domain becomes available for registration by anybody.
In the rare event that you do forget to renew your domain name and someone else renews it, there is very little you can do to recover it. In some cases, the new registrant may be willing to sell the domain.