Yes, grit is usually a mix of stones, calcium, and other minerals pigeons need for aid in digestion. The grinding actually takes place in the gizzard, which is a very thick and tough muscle, so that the rocks/sharp seeds don't hurt them.
If pigeons go without grit, they are still able to digest their food, but it takes more work and is more stressful to the system. Some seeds and things may not be fully broken down unless they are able to get to some source of grit (either provided by you or picked up off the ground).

 

Thanks for the info! I forgot about the gizzard, funny because that was the name of a little lizard I had years ago.

 

huh ha..gizzard.. name of a lizard.. funny indeed.