Where to cut my power supply for Arduino?
That thing is a Ferrite Filter. It's there to dampen Electromagnetic Interference (EMI) or high frequency noise from your power lines from interfering with your circuit. If you're not doing audio (or radio) work or you're not picky you can cut it off. If you are and it gets in the way, you can cut it off and add a ferrite bead elsewhere in your circuit.

Better than cutting it would be to add a proper plug to your circuit, but you've probably already considered that. If you're using a breadboard, this type of terminal block works well: http://search.digikey.com/scripts/DkSearch/dksus.dll?Detail&amp;name=ED2580-ND

A bit of warning: That looks like a laptop power supply, so it's probably capable of sourcing a fair amount of current. It's a little more powerful than a typical "wall wart". Be careful not to short anything, and don't work on your project while it's powered up.