What kind of DNS are you attempting?
There are two kinds of DNS.
There is authoratative DNS. This is the kind of DNS you use when you host your own domain and you want control over your domain. There are a lot of things to consider when doing this. Are you hosting your own email as well? Have you changed the NS records with your domain's registrar to make it take effect. If you don't know about these types of things, you probably shouldn't be hosting your own authoratative DNS. Leave it to someone who knows how DNS works (like your ISP or web host). I don't mean to be rude, but you can really shoot yourself (and possibly other people, too) in the foot if you do it wrong.
There is recursive DNS. This is DNS that handles simple resolution for hostnames on the internet. When you are setting up your IP stack, and you enter your DNS settings, this is the type of DNS it is referring to. Running your own recursive DNS can actually be very rewarding and can also speed up your internet activity significantly, especially if your ISP's servers are slow as molases. I used to have recursive DNS running on an old 486 that was infinitely faster than the DNS my ISP was running at the time. Best of all, recursive DNS will, at worst, prevent you from browsing the internet; it won't prevent other people from visiting any web site you might be running.
To answer your question, the serial number in the SOA record (which stands for Start Of Authority) is like a version number for your DNS (like acid_kewpie said). If you are running authoratative DNS, and you have a master DNS and slave DNS servers, it signals your slave DNS servers that there has been a change in your DNS records, so your slave DNS servers know that they need to get a new copy of the domain record for your domain from the master server. I think it might also allow recursive DNS servers know if there has been a change in the DNS records of a given domain so that, if a record has expired but the serial number has not changed, it doesn't necessarily have to go back and make a query to the authoratative DNS servers. It's basically a mechanism to cut down on unnecessary DNS traffic and load on authoratative DNS servers. It is for this reason that, if you make a change to your DNS, you should ALWAYS increment the serial number. The common convention for serial numbers is YYYYMMDDXX, where YYYY is the year, MM is the month, DD is the day, and XX is a sequence number. So if today is Feb 27, 2007, and it's the first modification I made that day, the serial number I would use would be 2007022700, and then if I made another change on that same day, it would then be 2007022701, etc.