Depends on your ISP and DHCP server, but if it is a Dynamic IP you can just disconnecting from the internet and reconnecting it to get assigned a new IP. I am guessing your using an 'always on' static IP so it won't change. If your IP is static, then you CAN’T change it without contacting your ISP or using a proxy.
Go to a site to check, for example:
http://www.whatismyip.com/You can go about manually assigning your own IP address. This IS possible, but you run a very high risk of your ISP banning you from connecting to the internet.
Also you have a MAC address, physical hardware address, assigned to each device while it can be spoofed, you can not actually change it. The DHCP server records your MAC and assigns an IP. If you disconnect from the ISP, you lose your IP address. The next time you connect, the DHCP server sees your MAC, looks to see if it has assigned an IP address to you before. If it has and the lease time has not expired, it will most likely give you the same IP address you had before disconnecting.
You can use proxy servers, can be annoying to find a good speed one and secure, but basically your computer connects to this server which forwards your internet requests on through a different IP making you a bit more anonymous.
