The first (and possibly most important) thing that happens when your domain expires is that it stops working. Not only are its records withdrawn from the registry's master DNS servers and replaced with "parking" records by the registrar, but its information becomes frozen and cannot be changed or updated.
After expiration, your domain enters the "renewal grace period." This is (usually) a 40-day period* during which the domain can be renewed at the regular cost.
After the renewal grace period, the domain enters the "redemption grace period," during which its status will be listed as "pending delete restorable." During This period, usually 30 days**, the domain can still be renewed, but it costs a lot more and takes a day or two to complete.
(Any renewal during either grace period will be effective as of the original expiration date; you can't get extra time by renewing after the domain expires.)
After the redemption grace period, the domain enters the "pending delete" status, where it typically remains for five days. It cannot be renewed during this period. After that, the domain is really, really gone and is free to be re-registered. Thus, the average domain can linger for as long as 75 days after deletion.
For awhile, it was the case that virtually every deleted name was instantly snapped up by speculators, but as of August 2008, that seems to be somewhat less common than it was. Nonetheless, the more valuable your domain is, the more crucial it is that you not take the chance.
*The length of the grace periods is not set by us and changes from time to time without notice to us. Thus we can't guarantee the lengths referenced here. Renew your domain as soon as you can if you care about it.
**Redemption grace period lengths also vary by TLD. For example, the .NAME domain currently provides only a five-day redemption grace period.
