Otaku is a Japanese term meaning "geek" or "nerd." It's generally viewed as insulting. You don't really want to be called one. It's not specific to anime or manga, it can refer to someone obsessed with anything. It's not just someone who likes anime. This term should not be confused with Hikkikomori or NEET, which refer to someone who shuts themselves off from society.
In the West, anime fans are often called Weeaboos. Weeaboos are essentially Japanophiles, obsessed with anime, manga, and Japanese culture. These Weeaboos find this term offensive, instead adopting the Japanese term Otaku for themselves. By calling themselves this, they make themselves even more of a Weeaboo, using a Japanese term to refer to identify themselves.
It's pointless arguing about this. Let's face it, we're all filthy Weeaboos. Embrace the term, don't reject it. If you don't want to call yourself a Weeaboo, don't call yourself an Otaku either. Simply refer to yourself as an "anime fan" or similar term. Otaku is incredibly cringey.
[spoiler]Otaku:
http://www.urbandictionary.com/define.php?term=otaku
https://en.m.wikipedia.org/wiki/Otaku
http://japaneselevelup.com/the-truth-behind-what-otaku-really-means/
Weeaboo:
http://www.urbandictionary.com/define.php?term=Weeaboo
http://knowyourmeme.com/memes/subcultures/weeaboo[/spoiler]