We could probably answer this if we knew which computer you had.
Get more information about your computer. Go to the Apple in the upper left corner of any window, then "About This Mac". Write down what it says about "version"and report that here. Now continue to "More Info..." Copy and paste the information here, but omit the serial number and Hardware UUID (if present).
Also tell us which version iTunes you are running.
Software update checks for updates but not complete upgrades for which you have to pay. I suspect you are using an older computer that needs upgrading (if it can).
Software update, upgrade--what's the difference? - http://support.apple.com/kb/HT1444
You still didn't provide full information. Look at http://www.everymac.com/systems/apple/imac/index-imac.html and you will see the number of iMacs that fit your description, all of which have widely varying capabilities.
So going on what I know:
1) Make sure your computer meets Snow Leopard technical specifications - http://support.apple.com/kb/SP575 This mostly means making sure it is an Intel Mac and has at least 1GB RAM (at least 2 GB is recommended). Go to the Apple in the upper left corner of any window to "About This Mac". Does it mention "PowerPC" (not Intel), or Intel? To check RAM continue to "More Info...". It should say you have at least 1GB of "memory".
2) Buy a Snow Leopard disc from Apple online
or telephone Apple. In the USA: (800) 676-2775 (contact information for other countries: http://support.apple.com/kb/HE57)
Alternatively you can buy from online retailers but anticipate paying much more than the original price which is the price still charged by Apple
3) Backing up your computer before upgrading is always a good idea.
4) After installing from the DVD, update your Snow Leopard installation using Software Update or manually downloading and installing the Mac OS X 10.6.8 Update Combo v.1.1 - http://support.apple.com/kb/DL1399
I have one of the first iMac 20inch with the bit built into the screen.
"Bit"? = big?
The truly first 20" iMacs will not upgrade to Snow Leopard which is required for iPad4 support.
Okay. Your computer is capable of running Snow Leopard (10.6.8) which is required for iTunes 10.7 or higher and which is required for your iPad4. Your best option right now would be to upgrade your operating system to Snow Leopard. I posted instructions earlier. It would be better if you had more RAM. At least 2GB, perhaps even 4 or more. You have the minimum required but that's like having a car which will just go at 50 mph so you're only just able to keep above the minimum on the freeway but isn't what you want for serious road trips. Also realize that Snow Leopard itself is pretty old (there's two newer system versions already) and you may want to do additional upgrades (which require Snow Leopard first so get it). Those would definitely require your computer have more RAM.
Once you have Snow Leopard and have it updated to 10.6.8 you can install iTunes. I recommend reviewing the hundreds if not thousands of posts on this forum over the past couple of days about iTunes 11. If you see something has disappeared in iTunes 11 which you use a lot you may not want to update iTunes that far and just stick with iTunes 10.7. You will have to not let your computer blindly do updating to prevent this. If you do want to stick with 10.7 here it is:
iTunes 10.7 for Mac - http://support.apple.com/kb/DL1576