I need to keep my internet on for a long time and I would like to save power during this time. So whenever I put my computer in standby mode, my internet gets disconnected. How can keep my internet on while the computer is on standby mode?

Do you manually activate the standby mode?

Simply put you CAN'T.
Standby can use just as much power as normal tick over, so there's really not much point in using it.
If you feel that you must leave your computer connected for long periods, noramlly for P2P programs, then standby is out of the question.
If you want to reduce the amount of power you are using while doing this then turn off your monitor and leave it at that.

If you don't want to switch off your monitor then you can activate the screen saver, which can reduce power consumption as well!

Ben11, sorry mate but that won't reduce power consumption at all!

Your monitor should have a built-in power save feature. (i.e. the power on LED will blink, etc.)