Using a channel relay
First of all, you'll soon blow up that LED. (Or the Arduino if you're really unlucky.) You need a current limiting resistor.

Secondly, most relays have two switch contacts: normally closed (N.C.) and normally open (N.O.). If you want your device to be off, unless the Arduino turns on the relay, use the N.O. connection. If you want your device to be on, unless the Arduino turns on the relay, use the N.C. connection.
It looks like you need N.O. 

N.b: 'closed' means on (i.e. the switch is closed and current is allowed to flow).

Finally, you seem to be using an external pull-up resistor for you switch, which is completely unnecessary: the Arduino has an internal pull-up resistor on every digital pin that you can enable using pinMode(pin, INPUT_PULLUP).

Edit: you didn't connect the relay to the button pin instead of the LED pin, did you?