Mac iBook G4 won't boot. When I hold down the Option key as it is booting, a blue screen appears. In the center of the screen is my hard drive icon with an x in the lower right hand corner. On the left and just below the icon is the refresh symbol and on the right, just below the icon is a straight arrow pointing right. What does this mean? How can I get past this screen to my desktop?
That symbol means your drive has crashed - or at least the system can no longer "see" the drive.
Do you have an install disk to boot from? If you do, use that, go into Disk Util and see if you can access your drive from there. If not, I would look at getting a replacement drive from eBay as your's is IDE/PATA and those are hard to come by new these days.
As @Allan said, your hard drive has a problem. You may be able to fix them with DiskWarrior and/or SpinRite. DiskWarrior will fix Mac-specific errors and will boot directly on your iBook (talk to their support to make sure they mail you something that you can boot from). SpinRite can fix drive-level data errors, but only runs on PCs (soon Intel Macs, which your iBook is not), so you'll need to completely remove the drive from the iBook and find a PC you can attach it to.
If you need to buy a new hard drive, these should work. The process for replacing the drive is here. It's not easy (I've done it), but is doable. (ifixit will happily sell you all the tools you need and a compatible hard drive.)
Allan's advice about finding the boot disk that came with your iBook (or another similar computer) is excellent, so I won't repeat it.
As to what you're seeing, you're in the boot-choice screen. (The refresh button tells the system to redo its search for bootable disks in case you plugged in an external disk or inserted a bootable CD.) The list of bootable discs show as the big square buttons and the rightward-pointing arrow tells the computer to attempt to boot the selected drive.