Towelroot promises to root every phone on the market which is older than June 3. Too be precise, which have a kernel compiled before June 3. It uses an exploit and provides you with an oneclick root process.
Simply download the Towelroot APK here: http://towelroot.com/tr.apk and install it.
The developer Geohot developed this for the AT&T and Verizon Galaxy S5, so they work 100%, but it also worked on The Galaxy S4, Galaxy S4 Active.
To get more information about devices witch are not fully working with Towelroot visit the official XDA Thread http://forum.xda-developers.com/showthread.php?t=2783157. You may also visit the website http://towelroot.com/.