Desaturate

Use the hue filter if you want to desaturate: (saturation = 0?)
ffmpeg -i input -vf hue=s=0 output
This is like using Colors → Saturation in the GIMP.

Grayscale

Use the format filter if you want to convert to grayscale format:
ffmpeg -i input -vf format=gray output
This is like using Image → Mode → Grayscale in the GIMP.