Checking the HP Printer Drivers
Before you go any further you might want to check this article on "Solving Windows 7 Print Spooler Problems". Check first if your Printer Spooler service is running and if it's not this service that causes the problem.
Once we are sure the printer is working, that we have a connection between the computer and the printer, and you still can't print, we will try to take a look at the software and printer drivers.
Each HP Printer comes with a CD or DVD with the printer software and printer drivers supported by your printer. Even if the Windows 7 detected your printer automatically and installed its built-in drivers, I always recommend installing the drivers provided by the manufacturer, in our case HP. Your Windows 7 might have the right drivers but they might be too old or depreciated to support all the functions of your new printer. Go on and insert the CD/DVD provided by the manufacturer and install the software and drivers provided. Follow the guide when connecting the printer as usually HP software is a bit tricky and will tell you to only connect the USB cable to the computer in the middle of the installation. Follow the guide exactly.
If you do not have the driver installation media available, you can always download and install the newest drivers from the HP website. Go to http://www.hp.com and select "Support and Drivers". On the next page select "Download drivers and software (and firmware)" and type in your printer model. Select the appropriate printer drivers, download and install them.
If you already have the drivers installed you can always replace them by selecting the "New Driver" option on the "Advanced" tab of Printer "Properties". After installing the newest printer drivers, try to print again.