Elementary Freya’s software center only lists Google Chromium browser and the not the widely popular Google Chrome browser for the simple reason being the later one is not a open source software.
Google Chromium vs Google Chrome
To adhere to FOSS rules, Google came up with the open source equivalent Chromium Browser which is identical to Chrome in terms of functionality, but the only drawback is it is not fully up-to-date in-par with Chrome. The current version of Chrome is Version 49.0.2623.87 (64-bit), where as Chromium is still in Version 48.0.2564.116.
On the other hand Google actively works on Chrome on daily basis for security fixes, bug releases, and speed optimization. So if you ask me, I would choose Google Chrome over Google Chromium browser anytime for my elementary OS. Note that Chrome is available for only 64-bit Linux distros only. So, if you have a 32-bit Linux distro, you should be installing Chromium instead. There is a bit of drawback when you install Chrome though – you will lose the capability of adding the minimize button because it has its own skin, unlike Chromium which uses native OS skin.
How to install Chromium Browser in elementary OS
If you are inclined to Chromium for some reason and would to install it, it is pretty straight forward and easiest installation. Launch ‘Software Center’ and search for Chromium. Typically, it is the first result ‘Chromium Web Browser’.
Go ahead and click on ‘Install’ and enter the root password when prompted and you are done with Chromium installation.