The meaning of a green arrow shown during a red light signal is virtually the same throughout the US but many states use different terminology to explain the green arrow. In New York the DMV defines a green arrow to mean that a driver may go in the direction of the arrow, but must yield the right-of-way to other traffic at the intersection as required by law.
In California the city of Pasadena says that a green arrow allows drivers to turn while the arrow is green but the driver must yield pedestrians, bicyclists and any vehicle in its path. The green arrow is what DOT engineers call a "protected" turn because oncoming traffic is stopped while the arrow is green.
Indiana and Minnesota use the same terminology to say that vehicular traffic facing the green arrow signal, shown alone or in a combination with another indication, may cautiously enter the intersection only to make the movement indicated by the arrow or other movement permitted by other indications at the same time.
Alternatively, a lit red arrow typically means no turn in that direction but many drivers turn after stopping or treat it like a regular red light.