The LFC stand for LHC File Catalog. It holds a mapping between a generic file name called Logical File Name (LFN) on to one or more physical locations of a file on the Grid. The LFN has a unix path-like representation. For example, a LFN like, /grid/myVO/mydiretcory/myfile can be mapped onto a physical location like srm://srm.grid.sara.nl/pnfs/grid.sara.nl/data/myvo/thisdirectory/thisfile.
There are two kinds of LFCs. A local LFC that runs at a specific site and holds references to files stored at that site and a global LFC that can hold references to files stored anywhere on the grid.