Many U.S. railroads adopted a system known as speed signaling, rather than the route signaling which is the norm in Britain.
Where there is a single signal head, as on long stretches of open track with no turnouts, the red, yellow & green indications mirror the equivalent British indications, except for the names, which are stop, approach, and clear respectively.
Once diverging routes come into the picture, the norm is for the signal to indicate the speed at which the train may travel for the route selected by a combination of two or more lights. The top head indicates the conditions for the full permissible speed of the line. The second head, below, then indicates the condition for medium speed. The definition for the latter can vary from one railroad to another, but typically limits speed to around 30 mph.
The signal will show green for whichever speed route is set and clear. So green over red indicates that it's clear to proceed at the maximum line speed. Red over green indicates that the full-speed route is impassable, but that the medium-speed route is clear. The indication is called medium clear.
Yellow then comes into play as an approach warning (the "caution" indication of U.K. signals) for each speed route. Yellow over red thus indicates approach for the full-speed route, just as for a yellow light alone. Red over yellow is medium approach, meaning that the full-speed route is impassable, but that the train may proceed at medium speed, expecting the next signal to be at stop.
It's possible for more than one light to be something other than red at a time. Yellow over green is called approach medium, meaning that the full-speed route will be red at the next signal, but that the medium-speed route is clear for at least two blocks ahead. In other words, the engineer may continue at full speed past this signal, but must slow to no greater than medium speed by the next signal.
A third signal head can be added at the bottom, which is used to indicate a route at which slow speed must not be exceeded. Again, the actual speed can vary, but is typically 15 mph. So green over red over red is clear; red over green over red is medium clear; red over red over green is slow clear. In other words, the top light shows the condition for full speed, the middle light for medium speed, and the bottom light for slow speed.
Again, a yellow light with the others red indicates approach, medium approach, or slow approach, expecting all red at the next signal. Yellow & green in combination indicates the speed at which an approach may be made to the next signal and the speed of the route which will be taken at that signal. So yellow over green over red is approach medium (the red indicating that no slow route is available). Red over yellow over green would be medium approach slow, meaning that the train may continue at medium speed past this signal, but must be down to slow speed for the next signal.
All lights red means stop, of course, since no route (full, medium, or slow speed) is clear.
Those are the basics of speed signaling. There are many extensions, such as lunar white to signal restricted speed, and flashing green or yellow for limited speed, which is higher than medium but below full speed.
Some U.S. railroads did adopt route signaling, which operates in a similar way to U.K. signaling, the top light indicating the main line, the next light down the diverging route, and so on. And some railroads adopted their own unique signals, like the position light signals of the old Pennsylvania R.R. or the position plus auxiliary lights of the B&O, which resemble no others.