Most DSL routers use a system called NAT (network address translation) to deal with communication between the computers on your network and the outside world. What this basically means is that the router (or modem, if using a separate ethernet modem) have one address that is visible to the internet, usually assigned automatically by your ISP when you connect, and one that is used by and visible to devices on your local network. This is done because every device connected to the internet needs a unique IP address, and if you wanted to connect two or more computers to the same DSL account at once you would need two or more IP addresses from your service provider. To get around this, only the modem/router is assigned the single IP address given to you by your service provider. The router than assigns completely different IP addresses to the computers connected to it, taken from a pool of reserved addresses (usually 192.168.x.x or 10.x.x.x). Call those "internal IP addresses". When one of the computers requests information from, say, Google, the router records the request to
www.google.com in a list called the NAT table, along with the internal IP address of the computer that requested it. When the Google server sends back its packets of data, the router looks up the source in its NAT table to see which computer (if any) requested that data, and forwards it to the appropriate internal IP address. In many respects this makes quite an effective firewall of sorts, since if, for example, another computer infected with a malicious program (like the infamous Blaster worm) randomly tries to attack your IP address, the incoming data will have no entry in the NAT table and the router will be unable to forward the data to any of the computers connected to it. In essence, the router only allows expected traffic in. It's not foolproof, of course, and it can cause problems with stuff like P2P software, or if you are hosting a service on your computer (web server, FTP, mail, etc.) because you can get incoming data from sources your computer didn't explicitly request all the time, but most routers also allow you to forward all incoming requests on certain ports to a particular internal IP address (this will be called "port redirection" or something similar in your router's config pages).
Anyway, I don't think you've got a problem here. Your external IP address, the address that is visible to the rest of the internet, is your 68.237.23.78 one, and thus is the one that whatismyip.com will report. The ones on your internal network must be falling into the 10.0.1.x range, which is within the block of reserved addresses for private networks. Basically, if things are working, you've got nothing to worry about.