The blue arrowhead

that swivels around the blue dot, tells you which way the phone's body is pointing, assuming that you are holding it with the screen horizontal. In portrait mode, that corresponds to a line running straight out from the top edge of the screen and parallel to the ground. This feature is available only in devices that have a compass sensor.
In other words, the blue arrowhead symbolizes a tiny phone drawn on the map's surface, oriented
on the map in the same way that the real phone is oriented
on the earth's surface.
If you want to confirm this statement, try this experiment for yourself (hold the device with the screen horizontal during this test). This experiment does not require that your compass be very accurate, only that it should be able to detect rotation reasonably well.
[1] Tap once or twice on the blue target (or compass icon) (near bottom right of map screen). The icon will toggle between a blue target like this :

.... and a blue compass like this ....

. This lets you select two corresponding modes of operation.
[2] Select the "blue compass" mode

as explained in step 1. In this mode, the red-and-grey compass

near the top left of the screen will be enabled and will point towards magnetic north as found by your compass sensor.
[3] Rotate the phone slowly in a horizontal plane while watching the blue arrowhead (center of screen..

). You will see that the arrowhead ALWAYS points to the TOP of the screen. The map itself swings around so that it's "north" is always aligned towards a certain physical direction. (That direction should be true north if your compass is any good, but that is not very important here). So if the device is held pointing east, then the arrowhead points to the map's "east" edge.
This observation may seem rather trivial or obvious, but it helps make sense of what comes next...
[4] Hold the device so that the TOP edge of the device's screen is PARALLEL to the red-and-grey compass. In other words, the phone's body is held horizontally and pointing EAST, based on the google map compass. (We don't care whether the compass is right or wrong). The whole thing should look like this:
[5] Now tap the blue compass icon so that it changes to blue target mode. The red-grey compass changes briefly into a map-style compass rose (Red arrowhead with an "N" below it), then it just disappears.
[6] See how the map swings through 90 degrees so that its north edge matches the top of the screen (which is the meaning and purpose of the "blue target" mode.
[7] Your phone is still pointing east (it really should, because did not move it while tapping the blue icon). But -- here's the important thing --the little arrow now points to the RIGHT edge of the screen, which is the map's EAST !!
Here is how it should look:
If you turn the device 45 degrees to point southeast then the little arrow will also swing to southeast on the map.
[8] Repeat steps 4 to 7 with any other orientation of the device, and you will soon be convinced that this, indeed, is what the arrow is all about.
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