You have a few possibilities - you need find out if there is updated HP software to operate the scanner (PowerPC apps are no longer supported in Lion or Mountain Lion).
Or you could try Image Capture in your Applications folder - that may work.
Or you could try the paid for application VueScan (www.hamrick.com) which is more fully-featured than Image Capture. You can use it in demo mode before committing to buy just to test it first.
I think you've got two problems going on here.
First, your driver's not working properly for some reason. A quick check of HP's web site shows they do offer a Mountain Lion driver package for this printer. I found it at this link:
Now, you didn't tell us how this printer's connected....do you have it connected via USB cable or are you connected via ethernet to your router? Setup via USB cable is generally very easy, as this printer is supported in Mountain Lion's list of supported devices.
I would install it by following the directions provided on HP's web site. However, if those directions aren't available, you should be able to unplug the printer's USB cable from your Mac. Then go to System Preferences --> Print and Scan. In there, if you see your printer listed, click on the "-" sign at the bottom of the left pane to delete it. Then plug the USB cable in and power the printer on. Go back to System Preferences --> Print and Scan, and this time click the "+" sign at the bottom of the left pane. The printer should show up in the list. Select it to install the drivers.
HP's web site is kinda spotty as to whether or not you'd be able to scan from the printer's controls or not...I suspect maybe not. And...scanning will most likely be limited to Image Capture. As Paul said, VueScan is a full-featured scanning engine, and worth it in my opinion. It's WAY better than Image Capture for scanning.
The other problem is one of compatibility with Mountain Lion...and I'm referring to the HP Image Edit application that gives you the PPC error message. This is because that's a (very) old program not written for Intel processors. It only runs on PowerPC processors, or on Intel processors with the Rosetta "translating" software. Apple stopped including Rosetta (most likely because they had to license it) back when they released the Lion operating system.
I don't know how old this printer is, but my guess is at least 5 years?