Your water meter is what registers the amount of water coming into your home. The main City water supply comes from the street to your water meter and then up to your house. The City maintains the main City water supply lines and your water meter up to the quest fitting which is on your side of the meter can. The maintenance of the water that comes after or out of the meter up to your home is your responsibility.
Your irrigation box is what houses the controls for your irrigation system. There is a double check valve inside the irrigation box that will allow you to turn off just the irrigation system. You maintain your irrigation system and the irrigation box. The city, as a courtesy, will shut off the double check valve if there is a major leak or water loss.