To make the discussion much simpler, let us concentrate on fuses that are used in most households. There are many types of fuses used in many different types of applications from protecting your computer's power supply to protecting heavy industrial equipment. Although varied in purposes, the same principles are employed.
The basic mechanism of the fuse is very easy to understand. When a device in your house develops a "short circuit" (meaning the current it normally uses to power it is now being diverted, because of a failure, to ground) it will conduct an excessive amount of current. The fuse that is protecting this outlet will pass that current but in doing so will get very hot and the metal element inside will melt, breaking the path of the current flow. Until this fuse is replaced with a new one, that outlet will not work.
Usually, fusible links are encased in an enclosure called the “fuse box” where the mains wiring comes in and is distributed to your house's outlets. The main fuse links your house to the power coming from the power company and then smaller individual fuses protect the wiring going to the networks of electric outlets in your home. Many applications today, such as household protection, are being replaced with electromechanical circuit breakers. These devices perform the same function as fuses but use a different mechanism to detect the excessive current flow and interrupt the circuit. They have the feature that once they trip and protect the circuit, they can be reset and used again, so you don't need keep a supply on hand as you do with fuses.