.CUE files are a file format that originated with CD writing software such as CDRWin. Back in the days when hard discs and processors were somewhat slower than they are today, it was considered sensible to create CD's in two stages. The first stage was to create an image file containing the raw data for the CD and an accompanying .CUE file to tell the CD writer what the image contained and then the second was to write the contents of the image to the CD, separating tracks where appropriate. The advantages of this approach will not be immediately apparent to the modern eye, but the basic reason was one of stability - CD writers are timing-critical devices and, in the days before they carried large memory buffers of their own, it was important that the host computer delivered data to them at a steady rate. Needless to say, a slow machine attempting to traipse around the hard disc to collate the necessary files was not a sensible solution and the .CUE/image file approach, when coupled with a freshly defragmented hard disc, prevented the generation of a large supply of useless coasters.
These days .CUE files are rarely used for CD creation, apart from with some open-source or Linux packages that still harken back to the 'good old days' when the simplest task was like a labour of Hercules and there was none of this point and click rubbish to make life easier. However, the file format has been adopted by MP3 manipulation programs and players as a handy way to track chapter points in MP3 files. By simply noting the time at which each chapter starts, a player can quickly skip to the relevant section of the MP3 file and begin to play. Audiotools can generate .CUE files for use in other programs, leaving your source file in one piece and simply recording the positions as track breaks.