At the Apple Icon at top left>About this Mac.
Then click on More Info>Hardware and report this upto but not including the Serial#...
Model Name: iMac
Model Identifier: iMac7,1
Processor Name: Intel Core 2 Duo
Processor Speed: 2.4 GHz
Number Of Processors: 1
Total Number Of Cores: 2
L2 Cache: 4 MB
Memory: 6 GB
Bus Speed: 800 MHz
Boot ROM Version: IM71.007A.B03
SMC Version (system): 1.21f4
Then you need to research all the problems people are having with the higher OSX versions, & make sure you have a bootable clone of what you have just in case.
then you must get 10.6 if they still have it, install it & update to 10.6.8 so you have the App Store to buy & download the huge 10.8 Installer.
Snow Leopard/10.6.x Requirements...
* Mac computer with an Intel processor
* 1GB of memory (I say 4GB at least, more if you can afford it)
* 5GB of available disk space (I say 30GB at least)
* DVD drive for installation
* Some features require a compatible Internet service provider; fees may apply.
* Some features require Apple’s MobileMe service; fees and terms apply.
Which apps work with Mac OS X 10.6?...
It's been pulled from the online store & Apple Stores, so you have to call Apple to buy it, last I heard.
Call Apple Sales...in the US: 1-800-MY-APPLE. Or Support... 1-800-275-2273
This article will tell you if you can update to 10.6 or later:
This article will tell you if you can update to 10.7 or later:
And this one will tell you if you can update to 10.8 or later: