What is the liquid often ladled onto the grill top in restaurants?
What you describe is grill basting - where a liquid is poured onto the grill and covered (with a metal salad bowl or a basting cover) to more evenly cook the food and maintain its moisture. The liquid is added towards the latter half of cooking, as adding it in the first half of cooking has no real beneficial effect. Additionally, the steam will melt cheese more evenly if grilling burger patties. 

Typically, this liquid is a stock (chicken or pork stock if cooking chicken, vegetable stock if cooking vegetables and beef stock if cooking beef) or water.

On a personal note, I sometimes dissolve a pinch of salt in the water if I don't have any stock available.