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(these instructions work for Windows XP. they should work the same on any Windows version, but you may have to poke around menus to find the correct option page)
just go into the control panel/network connections, right click on your internet connection icon, select properties. select the option for Internet Protocol (TCP/IP). this'll bring up the ip configuration screen.
it's most likely set to "Obtain an IP address automatically". this means your ISP gets to decide what IP address you use, and they're the only ones who get to decide when to change it. since you don't want this, just click "Use the following IP address". this lets you manually configure your own IP address. you can change this as many times as you want.
the IP address consists of four fields. these each consist of a number between 0 and 255. in some cases, '0', '1' and '255' have special meanings, so you're best off selecting a number between 2 and 254 inclusive for this field. pick four numbers to represent the new ip address you want, and enter one in each field.
Subnet Mask, this should always be set to 255.255.255.0 unless you know what you're doing.
for the Default Gateway, you would take the first three numbers you selected for your IP address, and replace the last number with a '1' (assuming you're using a 255.255.255.0 subnet mask like the last step recommended). for example, if your new IP address is "42.168.22.107", your default gateway should be set to "42.168.22.1"
There are also two fields for DNS servers. these fields are optional -- they're only used to find the address for a webpage or something if you don't already know the address. if you feel you need to set them, you'll have to find a DNS server which will accept connections from your new IP address. personally, i use "192.168.0.1" and it's never given me trouble, no matter what IP i set my computer to.
Note that the internet won't allow two people on at once if they both have the same IP address. if your new IP address doesn't let you connect to the internet, you will just have to try another one. there really isn't any way around this restriction, which is why most ISPs will try and give you your IP address automatically. there's plenty of unused addresses out there though, so trial and error will find you something available in short order.