There are a couple of things that can happen when you download an update. Sometimes the update is done automatically to the program, like iTunes, and the desktop icon is just a temporary element that came along with the download. Sometimes downloading an update places an icon on your desktop that requires that you double-click it and proceed with an actual update. Before you discard or double-click the desktop icon, open iTunes and in the upper-right corner under the Apple logo, click and open "About iTunes." This will show you your current version. If you are still at the old version of iTunes, you may need to now double-click the desktop icon to actually proceed with the update. But this isn't typical of Apple. Most of their updates load automatically and the desktop icon is temporary and you can trash it. If you need to manually update iTunes, you can trash the icon after updating.
Be careful about updates. It can be a viscious circle. If you update iTunes, you may need to update Quicktime too (which iTunes uses). Under the upper-left corner of the screen under the Apple logo, you can choose how you are warned about updates. The Mac comes default to check once a month, but you can choose "Never" or "Manually".
ok I just got my macbook pro on Monday. This is my first Mac ever, so needless to say I am a mac noob. I know windows really well, but not so much when it comes to mac and I dont want to mess anything up on my precious new computer...So anways when I booted it up for the first time and opened itunes it said there was an update avaliable for itunes so of course I said yes, to download. But for some strange reason it downloaded to my desktop! Now of course I can't move it to my applications folder and its stuck there on my desktop. It messes up the whole flow of the minimalist desktop! i'm guessing the only way to get rid of it is to trash it and then update itunes again? I mean there most likely isn't any way to just move it right?

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