This part is called the "pylon" or "strut" for the engine. It holds the engine on the wing, provides a path for all of the engine systems to connect, and includes the aerodynamic fairing to cover it all. The strut is designed to prevent a fire in the engine area from spreading to the wing. Mounting the engine below and forward of the wing leading edge provides some advantages over other options.
Once the strut is installed on the wing, it is intended to be permanently attached there. As shown in falstro's picture, the engine mounts are installed on the engine, and allow engines to be installed and removed as needed. When no engines are installed, a weight is suspended from the strut to balance the aircraft and prevent the plane from tipping backwards.
There are differences in design between manufacturers. Boeing struts are designed to break away from the wing under extreme loads. The strut attaches to the wing using fuse pins, which will fail at a specific loading, lower than the strength of the wing. This prevents the wing from breaking under the load, which could cause the fuel tanks to break open and start a fire. Examples of engines breaking away as designed are the Asiana 214 and Lion Air 904 incidents last year.
Airbus struts are designed to remain attached to the wing. This actually allows the structure to be less complex, and Airbus struts tend to be noticeably more narrow than comparable Boeing struts. Other methods are used to maintain safety in a crash, such as keeping the area directly above the engine as a "dry bay", which is not used to store fuel.
Here is an Airbus patent for a strut design using composite panels, which describes the strut.
As mentioned by falstro, there are multiple incidents where the structure in the pylon has failed, resulting in the engine coming off of the plane in flight. There have been both design and maintenance causes for such incidents, and both design and maintenance changes have been made to address those issues.