Seemingly simple question, but I'm afraid the answer will be a bit complicated: if you're on the Internet, you likely have two, not one, IP addresses, and the answer depends on which of these addresses you want to change. Furthermore: Either one of these two addresses could be assigned "statically" or "dynamically". And again, the answer depends on the assignment.
Taking this one by one:
(1) do you have one or two IP addresses?
If you are connected to broadband Internet via DSL or cable modem, you always have two IP addresses: an "internal" one (administered by your cable modem, DSL modem, wireless router, or whatever device you have hooked up that manages your "mini network". These devices always work with a limited number of addresses (often 24 or 48) within a particular number range (often "192.168.xxx.xxx" or "10.xxx.xxx.xxx").
If you are connected to the Internet via a professionally managed Local Area Network (LAN), then you also have two addresses, and internal and an external one. The difference is that the internal address range simply is much larger, and the rules to obtain addreses are enforced much more strictly.
(2) Do you have a static or a dynamically assigned address?
Easiest way to find that out is the following (assuming you have a Windows computer): go to the "Start" menu, select "Run", type in "cmd" and press Enter. This should give you a "DOS Window" (also called "Command Prompt"). In the DOS Window type "ipconfig /all" and press Enter.
The result is a screen full of network data that appears overwhelming at first, but is actually not that difficult to understand. Look for a line that reads "DHCP Enabled". If this is set on "Yes", you have a dynamically assigned address. If it's set on "No", you have a static address.
By the way: this screen also shows you both your internal and external IP address.
Do not close this window just yet!
Now, which address do you want to change?
(3) I want to change the internal address
If you are in a home environment, then the best way to change the IP is to simply set a new address for your computer yourself. The only watchout is that you need to select an address that
(a) nobody else is using, and
(b) is within the "legal" range.
For (a), the easiest way is to just pick an address you want to have, go into the DOS window we opened above, and "ping" the address. Let's say you want to have address "192.168.1.15". You would go into the DOS window, type "ping 192.168.1.15" and press Enter. If you get an error message ("no such device"), then this address is unused. If the ping command finds another device with that address, the address is already in use and you should try to find another, unused, address.
There are systematic ways to find out (b), but that would make the answer too complicated. Just try to find an unused address that is not very different from your current address. If your current address is "192.168.1.14", try "192.168.1.13" or "192.168.1.15". Chances are high these are legal addresses.
Now you can close the DOS window. Type "exit" and press "Enter".
OK, so now you know a legal address you want to change computer to. How do you do it? Open the Control Panel, open the "Connections" icon, and look for a connection named something like "Local Area Connection". Right click on it, and select "Properties". The Properties window that opens contains a section with the title "This connection uses the following items:". Find this section, and scroll down to find an entry "Internet Protocol (TCP/IP)". Click on it once to highlight it, and then press the "Properties" button. This opens up a window called "Internet Protocol (TCP/IP) Properties". On the "General" tab of this window, you can instruct Windows to "Use the following IP address:". Type in the legal, unused address you found by doing the above, and press "OK" until you're back at Windows.
If you are working in a professionally managed LAN, you will have to ask your network administrator to give you a new IP address. You can try the above steps, but well managed LANs won't permit users to select their own addresses. However, you won't break anything by trying, so give it a shot!
By the way: most of the methods given by respondents before me won't help you to get a new address. The programs assigning IP addresses try to make your address "semi-permanent". They recognize your computer when you log on. They will look up which address you had the last time you were on the network, and if this address is still available, they will give it to you.
(4) you want to change your external address
If you are connected to the Internet from a home environment (e.g. via DSL or cable modem), you will have to call your ISP to get a new external address. The ISP basically gives an IP address to your cable modem (or DSL modem) when this device goes online. And again, these addresses are "semi-permanent". They are typically reserved to you for 3-4 days. Unless you are willing to stay off-line for a week or so, you will always get the same address. Nothing you can do about that other than calling the ISP.
If you are connected via a professional LAN, then the public IP address you use is something your company, or your university, or your school (or whoever operates your LAN) has rented. You could of course talk to your network administrator, but changing a public facing IP address for a company or a bigger organization is no small undertaking. It can be done, but it's not free. They will want to know a pretty good explanation why you want to have a different address.
Hope this helps!