A 'band' refers to a single layer for a possibly multi-layer file. Most RasterLayer objects will refer to files with a single layer.
The term 'band' is frequently used in remote sensing to refer to a variable (layer) in a multi-variable dataset as these variables typically reperesent reflection in different bandwidths in the electromagnetic spectrum. But in that context, bands could be stored in a single or in separate files. In the context of the raster package, the term band is equivalent to a layer in a raster file.
nbands returns the number of bands of the file that a RasterLayer points to (and 1 if it does not point at any file). This functions also works for a RasterStack for which it is equivalent to
band returns the specific band the RasterLayer refers to (1 if the RasterLayer points at single layer file or does not point at any file).