The Mediterranean Bay Laurel (Laurus nobilis), known in Italy as alloro or lauro, is a common flavoring for Mediterranean meats and fish (also used to decorate the heads of victors). It's sometimes called True Bay, True Laurel, or Sweet Bay. Most Italians have one growing near their house, for cooking (and for hedges). It's used mostly fresh, sometimes slightly dried or dried, and is evergreen.
It is not related at all to the California Bay (Umbullularia californica), which is the common supermarket or farmer's market "Bay Leaf" in the US.
The flavor/fragrance profiles are quite different, as these commenters note. In my view, each is good but definitely different. To my American taste, the Med True Bay has a foreign, kerosene-like flavor which takes a little - but not much - getting used to.
Here's how Italians use Laurus nobilis. The excellent Tuscan steaks are always cooked with it, and grilled meats in general.