There are a few VNC programs. I use UltraVNC, but they all work the same.
On the machine TO BE controlled, run VNC Server, on the machine doing the controlling, run VNC VIEWER.
I asume you have a router between the computers and the cable modem or network device, so you have some sort of hardware firewall.
You will have to setup PORT FORWARDING on your router, and different routers do it differently. On your brother's router, you'll need to know his local IP address, and forward the VNC port 5900 to his local IP address.
On his server, he will pick a password for you to use.
Now, from his machine, he can go to http:// findmyip.com and it will display the public IP of the machine (WAN port of the router). That is the IP address you will use when running the VNC Viewer, on the viewer enter his IP address : 5900 e.g. 163.12.1.134:5900
If you get a prompt for a password, that will be from his VNC Server, all you have to do is get it right and voila, you're able to do stuff on his system.
There are many ways to do this, check out logmein.com they offer "free" accounts. Also, if your router supports DynDNS, you can create an account and the router will always be reachable even if the ISP changes the WAN IP address if you setup the router to use Dynamic DNS.
I notice two other recommendations for logmein.com. I agree, it is a great program, but I'm not so sure about the security issues since they know a lot about the computers working through their system.
As to changing IP addresses, that's the purpose of DYNDNS, go to www.dyndns.com and learn about it. Most newer routers allow you to setup a dyndns name that you've preregistered with dyndns.com. For example, I use an address like (but not exactly) "hal9000.somedomain.com" and it always has the correct public IP for my system.
With dyndns, the dynamic nature of the IP address becomes a moot point. This is basically what logmein.com is doing, but they have a nice client/server program. Again, my concern is who knows about my computer, and where is this data being kept?