Chassis mount devices are designed to mount onto solid surfaces, typically the back of cabinets. They tend to be larger, higher powered devices designed for usage in industrial environments.
DIN rails, named after the German institute that established the standards, is a standard mounting method used for devices in industrial cabinets. The standard mounting and size simplifies and cleans up the organization of cabinets.
Through hole mounting is specific to small devices that are going to mounted either on a PCB or a breadboard. These devices are designed for their leads to pass through the PCB and be soldered on the other side. These tend to be larger than their surface mount counterparts, with higher inductances, but are easier to deal with and are more mechanically sound.
Surface mounting is an increasingly popular way of mounting devices onto PCBs. With surface mount devices, the leads stay on, and are soldered on, the same side of the board as the device. By keeping the soldering on one side, less holes need to be made in the PCB, meaning smaller sizes can be achieved. While they save space and reduce inductance, certain types of surface mount devices are difficult to hand solder and others are, for all practical purposes, impossible to hand solder.
Panel mounting is typically for larger components that are used in electrical cabinets. These are designed to be mounted into the sides or front of the cabinet but can, in certain circumstances, be mounted in the back of the cabinet similar to chassis mount.
With connectors, cable mounting is very common as this is the way that cables are terminated in a way that allow them to either be connected to other cables or to devices. Cable mounting can oftentimes require special tools for termination but these tools typically yield professional quality connections.
Essentially, rack mounting is a method of housing electrical equipment, or other bulky unmanageable objects, within a rigid frame. This frame, or rack, will generally contain multiple bay levels, where the equipment can be placed. Additionally, equipment can be directly mounted to the rails of the frame and secured in place with screws.