You're not doing anything wrong, what's happening is your download is corrupted so when OS X does its integrity check, its' failing it. This is the message you are seeing.
It would be very helpful if you could post exactly which computer you are using, specifically the Model Identifier:, e.g MacBookPro10,1 and or MacBook Pro (Retina, 15-inch, Early 2013). If your computer can run the newest software (El Capitan), then you just need to open the mac app store, hit 'Get' and follow the steps on the screen. El Capitan is a free upgrade!
If your computer is too old to upgrade past 10.7 (Lion), you have some options as well:
- Download the standalone updater: OS X Lion Update 10.7.5 (Client Combo)
- Reboot, and try again a few times. Sometimes it gets through
- Try from a friend's house who uses a different ISP. I've seen this happen when an ISP's CDN caches a corrupted copy.
Hopefully you can just upgrade to El Capitan and skip this whole thing.