Can I use an external hard drive with an Android phone? For years I've been downloading things onto my Android then transferring them to SD cards via the usb port connected to a card-reader. 

Could I use a basic unpowered external hard drive instead, such as a Sonnics? (Phone is not rooted)
It is possible but not with all devices


USB OTG (USB On the Go) capability of the device both from the hardware and ROM (specifically kernel) decides whether your device is capable of connecting to a hard drive. 
You would need to search on Internet if your device supports it or not. There are apps on Play Store, that can tell you but they require root (usually )
Rooting is not required if your device supports OTG connection to hard drive. Rooting helps in flashing a custom kernel that can enable OTG , provided your hardware supports and your device is supported by developers who make custom ROMs and kernels (popular devices are supported (search on XDA for your specific device model)
Also be aware that certain devices support OTG capability in terms of connecting a flash drive but not external hard disc (powered externally or not )


For more, see usb-on-the-go ( tag added to the question ) and frequent questions and How can I determine if my device has USB Host Mode (OTG) support?