UNO R3 Development Board ATmega328P ATmega16U2 with USB cable for Arduino

Is “http://www.amazon.in/Development-Board-ATmega328P-ATmega16U2-Arduino/dp/B00H1HR576/ref=sr_1_1?ie=UTF8&qid=1418280690&sr=8-1” good for starters?
I am a starter to hardware but I know c++ and c. Does this board run on battery?

Yep it's ideal to start with.

It has a USB-B socket, like a printer. You can usually power it from there even if you aren't programming.

Otherwise yes, a battery, but there isn't one built in. Or of course an ac/dc wall adapter.

View the official product page here.

Will the ac adapter be included in the package?

Saarisht: Will the ac adapter be included in the package?

You would need to check that with your supplier; probably not, and that link you gave shows just the usb cable.

What's the difference between fake and real arduino board?

It isn't a fake, it just doesn't claim to be made by Arduino, it is a clone, as far as I can tell.