Arduino water pump with relay not working
The relay you have is rated at 12VDC. You need 12V to power the relay you are using when your Arduino can only output 5V max. You probably bought the wrong relay, because the one you linked on eBay which is the right one is different to the one you have.

I can tell because in the photo, your relay is marked as SRD-12VDC-SL-C:


And in the datasheet below...
https://www.ghielectronics.com/downloads/man/20084141716341001RelayX1.pdf
That 12VDC refers to the voltage needed on the coil to activate the relay. So buy a new relay that has a coil rated at 5V or refund and replace the relay with the SRD-5VDC-SL-C if they gave you the wrong product.