What is this and do I call plumber or the city?

Looks like something digging in the yard. The city will ask if your meter spins when your faucets are off. If it's spinning it's your problem and have to call a plumber.

Those appear to be holes that filled with water. Is it only wet there or is it coming from up the hill?

If it's initiated there, the hole on the left looks like a sprinkler or main box.

Front yard?? Really? I always take a leak in the backyard.

Maybe I'm blind but I can't tell what I'm looking at. Maybe a picture during daytime would help.

Leaking sprinkler head?

Open the valve box and see where the extra water is coming from, right or left, then figure out if that pipe goes to your house or comes to your house.

If it goes to your house it's your problem, if it comes to your house it's your water suppliers problem, city/MUD/etc.

As craig09 said it looks like a mainline leak downstream from the double check valve and possibly at the master valve. Turn off one of the handles on the double check and then pump out both boxes and then turn double check back on and see if you can see any leaks. You can also dig out around the master valve and then turn on the water.
Where is your meter located?