The best bet is, if at all possible, to contact the original owner to have him pass over the control of the domain. The next best option is to work with the registrar, who may be able to work with you to contact the account owner, or they may have some other ability to help you. The last, and worst option, is to wait for it to completely expire.
Once a domain name expires, it is given a grace period. As the site is now not available, that may be a sign that this grace period has passed. After the grace period, there is a "redemption" period. This period of time allows the registrar to give the owner a little more time to redeem the domain, but they will often charge a heavy fee to retrieve a domain name from redemption. This fee is often around $100. This is still better, if it is available, because it is the last chance you have to be sure to acquire the domain.
Once the redemption period has passed, the registrar will release the domain. In a perfect world, you would just go register it at that time. However, there are many scavenger buyers who watch for expiring domains and snatch them up as they come back on the market. They don't do this because they have any desire to use them, but because they feel confident that if somebody had wanted them before, somebody else will want them again. Sadly, there is currently no law against this, and these organizations will often charge large amounts of money, often in the thousands of dollars. Even if you do decide that this sort of investment is worth it for you, there's still the chance that somebody else thinks it's worth it for them, even if for nothing more than capitalizing on the SEO benefits. So, it's not a guarantee.
Therefore, if you simply can't gain access to the domain through the named registrant, then you're probably better off looking for some new domain and start working toward moving away from it. That means updating your links and email addresses, etc. With proper planning you may be able to minimize the impact of a new domain name. Also, remember that there are many TLDs out there now. So, you might be able to find a similar name with a different TLD. When you do, next time, be sure to leave your own information on it as the registrar, and maybe consider auto-renewal. I've had to learn the same lesson, myself, and it is not fun.