…do you mean that the i Mac was bought brand new (i.e., the newest model available in stores in the 2009-2010 time frame), or did you potentially buy an older model (i.e., from 2007-2008) in the 2009-2010 time frame?
If you are sure you bought it brand new in stores between 2009-2010, it was probably what Wikipedia calls the Early 2009 model.
Even if all of the previous deductions were incorrect, as long as you are 100% certain that the i Mac has an aluminum housing (i.e., the outside is gray, and not white), then even in the worst-case scenario in which your i Mac is from Mid 2007, you are guaranteed to be able to install OS X 9 (Mavericks).
So rest assured, you can upgrade all the way up to Mavericks if you so desire. You can up-grade from 10.5.8 to 10.6 by purchasing the installation DVD for .99 (w/ free shipping), then up-date 10.6 to 10.6.8, for free using the Apple Software Update application already installed on your i Mac.
A friend of mine has a desktop Apple computer (I googled Apple computer images and it looks like an i Mac). When we tried to upgrade it the OS reported that there are no updates available.
We googled around to find alternative ways of upgrading the system - 3-4 different clicking paths all leading to the same system update program - but with same outcome.
Finally we tried to download it but we could not find the 10.6 upgrade anywhere.
The system does not allow to upgrade to any version higher than 10.6. From there, if the machine meets the requirements for 10.6 the process is to run the installer and replace / upgrade any peripherals and software that no longer work.
I'm not able to check it now, but I remember for sure that the computer was bought around 2009-2010, and it is one of these two (1, 2), which should mean that it can be upgraded even up to Maverick. You can use System Profiler to find to Model ID of your friend's i Mac (see this KB article) -- that'll tell you for sure which generation of i Mac you have, and whether it can run Mavericks (i Mac7,1 and later). From there, all upgrades come electronically from Apple.
Again, if the hardware is compatible, you can jump to Mavericks to free (though unlikely for most 10.5 era hardware).
Apple also sells Lion and Mountain Lion from the web store if you need an upgrade that isn't the latest and greatest.
Your first practical step is to go to and find in support the article on indentifying your Mac to determine the hardware capabilities before spending any time or money on incompatible upgrades.