Arduino I/O Shield relay won't switch
Unless you share a schematic, it will be very difficult to provide a good answer. 

Here are the things that you might be doing wrong:

1) Using a power supply which is not able to provide enough current to the relay.

2) Using a power supply of wrong voltage as required by relay coil. The relay might be needing 12 V and you might be trying to power it up with a 5 V power supply.

3) You might be trying to drive the relay directly with your GPIO pin (without using any external transistor switch).

4) Trying to raise a dead relay. There is a possibility that the relay is damaged. Check it by powering it up directly using a battery or another adequate power supply that you got.