Grass is primarily cellulose. It's difficult for most animals to break down into usable nutrients, but ruminants, like cows, have multiple stomachs that act as fermentors/bioreactors, where specialized bacteria break down cellulose in a way that provides nutrition for the cow.

Grain is primarily carbohydrate and protein, which is why it's sought after by animals that lack ruminant guts.

When cows are fed on primarily grass, it means a) they're usually out wandering around a pasture and b) they're eating their typical diet. This tends to produce lean meat and mean that the cows weren't primarily crammed in a feedlot.

When cows are fed on primarily grain, the energy-density of their diet is much higher, and it takes much less effort for them to break down, so they tend to put on more fat. "Grain-fed" typically means they were raised on a mixed diet and "finished" on grain (fed a primarily grain-based, energy-dense diet toward the end of their lives), which results in fattier, marbled cuts of meat that may be more flavorful/tender.