External hard drives are internal hard drives, typically SATA, that have been put into an enclosure so that they can be used outside of a computer and are often quite portable.
Laptop drives that have been put into an enclosure typically are bus-powered, which means they don't need an external power supply. Larger desktop drives are cheaper but require an external power supply.
Various connectors are used, from the latest connectors (USB 3.0 and Thunderbolt) to older connectors (USB 2.0, FireWire 400 & 800, eSATA). An external drive may contain one or more connectivity options.