You have an iPod and an iTouch. Your iPod used to become updated on a different computer. How do I enable that new computer to update that iPod as well as an iTouch you ask?
Right now your songs on it show up "greyed out" and you can't edit or move them around. You have iTunes on this computer but can only update your iTouch on it. If you are transferring that iPod from one computer to another that runs same operating system ie. from Windows to Windows then it shouldn't be too tough a process.
If you are trying to transfer that iPod from different operating systems ie. from Windows to Mac then that will be more difficult. I'm not exactly sure if you just want to move that device and sync or transfer files from that iPod to that different computer, so please read Apple's using iPod with multiple computers article. You can manually add items to your Apple iPod and or iTouch.
If you can't manually add items to your Apple device, then there are some additional steps required. Hopefully, you are now able to use your ipod with multiple computers.