This is a frequently asked question and hasn't been answered a lot, so I will make a tutorial on how to do it!
Requirements:
A Name Tag (duh)
An Anvil
A mob to name

1. First off, you will need to find your name tag, either in a dungeon, stronghold, temple, or abandoned mineshaft. They are often (always) titled "Name Tag".

2. You must craft an anvil, if you don't know how, check out this page!

3. Place an anvil, where ever you want!

4. Get some XP from mining or killing mobs

5. Take your name tag to the anvil and open the anvil GUI (just right click the anvil)

6. Place the Name Tag in the Left box and here should be the title "Name Tag" at the top.

7. Edit the Text on the top, to whatever you want (Bobby and Hans are good names)

8. Select the Name Tag in the right box and remove it from the Anvil.

9. Approach your mob that you want to name.

10. (OPTIONAL) Do a little dance

11. Right click your mob to name it!

12. Enjoy!

Hope I could help!