Wow - an Arduino newbie having read enough to use active LOW logic like they should! That's good!
You need a pullup for the button pin. If you do pinMode(13,INPUT_PULLUP) it should work - assuming it is wired correctly. Otherwise, the input is "floating" and will transition randomly when the button isn't pressed as it picks up noise from the environment. One can also use an external resistor instead, but this is rarely necessary, since there are those internal pullups. If you ever find yourself needing an external pullup or pulldown resistor, 4.7k-10k is a good "default" value.
Is it wired correctly? Button should connect to ground when pressed. With those el-cheapo 4-pin buttons that are common in starter kits, there are two pairs of pins that are each connected internally - so if you use the wrong two pins, they'll always be connected.
Finally - pin13 is the pin used for the builtin LED
On an Uno/Leonardo (and some of the other boards - I don't use the non-avr boards, so I don't know off hand which), pin 13 is "buffered" to drive the LED - there's a separate component (specifically, an op amp wired as a comparator) which watches the pin without loading it down, so the LED doesn't put any load on pin 13. So it should be find to use them for the button on those boards.
However, if you're using a nano, pro mini, or micro - smaller, cheaper boards - these do not have that, and have the LED connected directly to pin13 through a resistor. This puts a load on the pin such that the internal pullup - or even a typical external pullup - won't be able to pull the pin high. On those boards, use a different pin for the button (an unusually strong external pullup would also work)