This weekend I mowed the yard. My neighbor says he cut the grass. Did I cut the grass, or maybe I mowed the lawn, or did I cut the yard?
When does one mow, and when does one cut? Is it grass, or yard, or lawn?
"Mow the lawn", "cut the grass", and "mow the yard" are all basically describing the same thing, though I'd say "mow the yard" is more common in US English and the other two in British English. The only one that sounds wrong is the fourth; I've not heard the phrase "cut the yard" used before.
I'd say you can mow and cut the grass, mow the yard, and mow the lawn.
In a survey of 10553 respondents in the United States, the favoured expressions were:
mow the lawn 66.79%
cut the grass 18.38%
mow the grass 5.75%
cut the lawn 0.64%
Geographically, most areas favoured mow the lawn
but cut the grass was preferred in a few localities, mostly southern states.
Mow means (or usually means) specifically to cut down (grass) with a machine. You could say "this weekend I did the mowing", with the understanding that you cut grass with a tool of some description.
As touched on above, lawn is simply regularly mown grass. You could expect a yard (or similar) to be regularly tended to, and thus contain lawn. In my experience saying cut the grass carries the connotation that what's there could hardly be referred to as lawn (possibly with a haughty raise of the nose).
"Mowing the lawn" is universally understood in American parlance. It refers to the action of cutting wide swaths or a large area of grass (the lawn) using a machine. "Cutting the grass" is also well-understood, and refers to the same action, optionally with the additional steps of trimming and edging (which also cut the grass, but are more precise than the connotation of "mowing" indicates).
In Britain, a lawn would generally be a well-kept area of domestic grass. You'd probably not describe an overgrown or partly bare area of domestic grass as a lawn except with a note or irony.
"Mow" can have a rather agricultural feel to it, so "mow the grass" is perhaps more what a farmer might do.