Quick guide to root Android phones

0. Google how to unlock the bootloader for the phone.
1. Download and install the universal adb drivers from https://developer.android.com/studio/releases/platform-tools.html. Alternatively, download them from https://adb.clockworkmod.com
2. To check if adb was correctly installed, follow instructions here.
2. Enable USB debugging on your phone.
3. Download the latest version of SuperSU and put it in the phone
4. Install TWRP using this guide http://androiding.how/how-to-install-twrp-recovery-via-fastboot/ or this guide http://www.howtogeek.com/240047/how-to-flash-twrp-recovery-on-your-android-phone/
5. Install SuperSU using TWRP

Android root related words

ROM: modified version of android
Flash: To install something on your device. To replace the OS that is installed on your phone.
Recovery: the software on the phone that lets you make backups, flash ROMS, and other system-level tasks.
TRWP: A custom recovery
SuperSU: A management tool app. It prevents and provides access to root facilities.

Sources: http://lifehacker.com/5789397/the-always-up-to-date-guide-to-rooting-any-android-phone
http://android.stackexchange.com/questions/6183/what-do-root-flash-and-rom-mean