The term "interface" can refer to either a hardware connection or a user interface. It can also be used as a verb, describing how two devices connect to each other.
A hardware interface is used to connect two or more electronic devices together. For example, a printer typically connects to a computer via a USB interface. Therefore, the USB port on the computer is considered the hardware interface. The printer itself also has a USB interface, which is where the other end of the USB cable connects. Several common peripherals connect to a computer via USB, while other devices use a Firewire connection or other interface. Ethernet connections are commonly used for networking, which is why most cable modems and routers have an Ethernet interface.
Many other electronic devices besides computers use some type of interface to connect to other equipment. For example, a TV may connect to a Blu-ray player via an HDMI cable and may connect to a cable box using component cables. Audio devices may have either analog or digital audio connections and may include a MIDI interface, which is used to transfer MIDI data. iPods have a proprietary "dock connector" interface, which allows them to connect to a power source and transfer data via USB.
Since there are many different types of electronic devices, there are also a lot of hardware interfaces. Fortunately, standards like USB, Firewire, HDMI, and MIDI have helped consolidate the number of interfaces into a manageable number. After all, it would be pretty difficult if each digital camera, printer, keyboard, and mouse used a different interface. Computers would need a lot more ports on the back!