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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.