In the United States, a traffic light showing a green left arrow indicates a "protected left turn": oncoming traffic has a red signal, as do conflicting pedestrian crossings. Here is an example, taken from an article by the Federal Highway Administration:

In Europe, a directional arrow signal does not mean that opposing traffic has a red signal. It just means that you can proceed in the direction indicated by the arrow, but still must yield the right of way as usual.
I was once in Italy, turning left at a 4-way intersection where traffic had to turn left or right because the street directly ahead was a one-way street in the wrong direction. To indicate this, the signal showed a left green arrow and a right green arrow. Unfortunately, I irritated an oncoming driver by cutting in front of him. He also irritated me because I thought he was running a red light. It was only a few moments later that I realized that I had been in the wrong, and why.
I can imagine that a European driver in the United States, facing a green left arrow, with cars approaching the intersection from the other direction, would probably wait before turning left, at least until it was clear that those cars were stopping. Any American drivers behind that European would most likely be irritated by that behavior.
Another difference with traffic signals: Most European signals are designed so it is not possible to see the signal for cross traffic, but this is possible with most American signals.
American intersections typically have at least one signal head on the far side of the intersection. Among other advantages, this allows the driver stopped at the stop line to see when the signal changes, because the signal is several meters ahead rather than directly overhead. See, for example, this image:

A consequence of this is that when you are waiting at a red light, you can see the signal for the cross traffic at a very oblique angle. You can usually see which lamp is illuminated by looking for the light reflecting from the sun shades. When the light for the cross traffic turns yellow, you can prepare to start driving.
European signals are generally placed only at the near side of the intersection. Some consequences of this difference:
- In Europe, when you stop your car at the stop line, you can't see the main signal head (unless you have a convertible or perhaps a sun roof). Most signals have a small signal head mounted low on the pole, so the driver of the first car can see when to go. (If you are stopped at a signal that lacks this second, smaller light, and you are tall like me, you have a very difficult time seeing the signal.) An example:

- If you allow your car to stop a little bit past the stop line, you are completely out of luck because you cannot see the signal at all.
- Many European countries have a signal phase immediately preceding the green light, to warn drivers to prepare to start driving. A couple of seconds before the light turns green, the yellow light turns on, so the signal is showing both red and yellow. This is not necessary in the US, because drivers can see when the signals are changing. An example:

- In some US intersections, of course, the light doesn't turn green immediately after the cross traffic gets a red light, because, for example, there might be a protected left turn for opposing traffic, or the intersection might be unusually complicated. In such cases, there is often a sign saying "DELAYED GREEN" or "DELAYED GREEN WAIT" in black letters on a white background. This is supposed to tell drivers not to start driving when the cross signal turns red, and to wait patiently for their own light to turn green.

When I was a kid, I had no idea why one might have a "delayed green" -- I always thought, "of course it's delayed; if the green doesn't wait for the other light to turn red, then what's the point of having a traffic light?"
Finally, the delayed green signal can be an arrow indicating a protected left turn. Here is an example from Harrisonburg, Virginia (see the hburgnews.com article for a discussion):

(The five-lamp signal has, at the top, a circular red light, with yellow and green arrows on the left, and yellow and green circles on the right.)
The US has the keep-your-lane system
I don't think this is true... and I know it's now not true in Kansas. You can now be ticketed for staying in the left lane.