Extracting one band from three band raster in ArcMap?

The Geoprocessing tool that will extract out a single band from your three band raster is the Make Raster layer tool. You would set the band index parameter to the band you wish to extract.

This is a quick and easy way to extract the band and as its a tool you could embed it into a models workflow. Note its not a permanent layer so you would need to save it off if you wanted to give it to another person.

If you double click on your raster when you add it, it will become possible to select a single band. You can then export this single band using "right click" export data, but this is not necessary.

This uses Geoprocessing rather than the ArcMap GUI but it has a bit more flexibility.

Use the Composite Band tool under Data Management->Raster->Raster Processing. Open the tool dialog and expand your raster in catalog tree (showing the individual raster bands). Now drag whatever bands you want into the Input Rasters box and name the output whatever you like. This tool can be used to extract any number of bands from a raster or to combine any number of bands.