First of all, have you ever been able to successfully print using this printer? If so, my first question would be, what has changed recently on the computer and/or printer that might have contributed to this problem? If you connected another device to the computer recently, it may somehow be interfering with your printer, either because of the physical hardware connection or because of the new device's drivers.
If you have never been able to print using this printer on this computer, the first thing I'd look for is what you looked for...connection problems. If everything is plugged in correctly, another issue might be a bad cord. Assuming the printer is powering up ok, you can rule out a bad power cord. If you are using a USB connector to attach the printer to the computer, check and see if the cord is damaged. It's very common to roll over a cord with your chair, damaging it. Even if it doesn't appear damaged, see if a replacement cord will work.
You could be having a problem with your drivers. You can try to uninstall and reinstall your printer on the computer. This will require the original driver disk that came with the printer. To uninstall the printer, you can follow the steps on this HP.com website:
Uninstalling Software in Windows
Of course, I'm assuming you are using Windows and probably Windows XP, since that's the most common choice, but I could be incorrect.
You can get all the uninstall and reinstall information on the following web page:
Setup and Install
Have you tried printing a test page? If not, see Step 2 of the instructions on the following page:
Printing a test page is a good way to see if the issue is with the printer or the computer/application. You might also want to try to print different documents created with different applications (Word, Excel, PowerPoint) to see if they all don't print or only docs from one application don't print.
Please reply with your results after you've given these troubleshooting steps a try. Good luck.