Pull down resistor for Arduino input to read LOW when no voltage from switch
Without the resistor, if you press a button, you'll short 5V to 0V and BAD THINGS will happen. 

On another hand, if we leave bad things for a moment as a thought experiment, what level would arduino read? 

5V ----+---- 0V

The voltage at the + would be defined by the ratio of resistances of wires going from 5V and 0V to a + point. This would be tricky to set up properly, so it's better to put a big resistor (much bigger than wire resistance) and have the result predictable.



Edit

by BAD THINGS I mean - what does happen when you short a battery? In theory unlimited current flows from positive terminal to negative, in reality this current is limited by internal resistance of the battery, battery heats up, potentially explodes or leaks. If you are powering from USB - usb circuitry inside computer fries (this rarely happens as computers usually have protection circuitry). If you are powering circuit from a wall-wart adapter, a fuse in it might blow or it can catch a fire if it's of some extremely crappy chinese origin. If your power supply is strong and robust enough (like lead acid battery or lithium cell), a power trace on a PCB can burn off. In this particular case nothing bad would happen - arduino has a resettable PTC fuse on board, it will just reboot.