It's more a term for movements started by the common folk, as opposed to those started by the likes of politicians, big buisiness, and others in positions of power. The right wing likes to use the term because it makes their followers feel like they're the ones in control, rising up and taking the power back from the "liberal elites."
In reality, of course, the Tea Party in particular and neo conservatism in general was started by and is lead by the rich and powerful. That's why some people like to call right wing "grass roots" movements "astroturf."