A fuse is a device that stops the flow of current once a current threshold is reached. In electronic devices these are usually used for overcurrent protection. In microeletronic devices these are typically used for configuration.

Example polysilicon fuses (National Semiconductor DMPAL16R):

Both the left and the bottom fuses are blown (non-conducting) but the upper right fuse is still conducting.

These are non-conductive until sufficient current flows through. See Wikipedia for details.