If someone has a domain name registered and it is going to expire, can I register the day after it expires providing that they do not renew it? Or is there some type of "waiting period"?

You have to wait.

You might want to take a look at snapnames.com and namewinner.com. They have better chance getting the domain before you. :cool:

Note that 'expiring' is a billing thing, and that is different than the domain 'dropping' and being available for anyone to register.