Just to the left of the Spotlight icon in the menubar is the icon which tells you which app is active. If you click on it, the menu will drop down showing which apps you can click on to open another app window.
Not on this Mac. The time (and date, hidden) occupy that spot. None of the other icons up there reveal the running apps either. I looked real quick for the preference panel(?) that lets you customize the menu, but didn't see it. I'll take more time to look later. 
Uhh, I'm not sure if it's changed on Leopard, but in Tiger and Panther, there are little black triangles either underneath or to the side of the application icons on the dock, depending on where your dock is located.
Not on this Mac. My Dock is on the right side (yours being on the wrong side) and hidden unless I request its appearance. No little triangles anywhere. 
For some reason many users never use one of the most useful app switching function built into the OS. Hold the command-key down and tap the tab key (don't hold the tab key down, but keep holding the command key). You will now see a list of all running apps. Tapping the tab key will now take you, sequentially, to each app, from left to right. Add the shift key, and you can go in the other direction. Stopping on any app's icon will bring that app "to the front." But when that app's icon is highlighted, at least the "Quit" command is available. Remember, you still have the command key down, so all you need to do is also press the "Q" key to Quit that app.
That is cool! I'll have to remember that one. 
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Of course, the Dock also shows what apps are running, if you make it visible.
And how is that? My Dock doesn't show any difference in Icons of apps that are running vs. icons that I put there to have quick access to. Other than the latter, when an app is opened, its icon appears in the Dock and I recognize it as a "visitor" rather than a "resident" icon.
Well, by gosh, I never used that one Jim, and just did it, and I that's much better than perusing the dock for the little black triangles!
That's why someone has to ask the basic questions around here. 
To really know what is currently running, one must access Applications/Utilities/Activity Monitor.
Not sure I need to know that, but just how does one "access" Activity Monitor? Go there and double click the icon? Really, I just want to see the ones I have/had running on purpose.
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can you put the Dock on the top?
Not in Leopard. And it required a third-party app even in Tiger.
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if that's the case, somebody screwed up
Agreed, if you are not using an Apple monitor which should be setup to use the whole but not more than the whole screen. Or a somewhat out-of-'whack' monitor.
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Couldn't you "minimize" the Activity Monitor to the Dock?
Not sure why not, clicking the yellow 'candy' dot does exactly that on my machine.
But you can also drag any app's icon to the dock. That makes them "resident."
But you can also drag any app's icon to the dock. That makes them "resident."
True. But my objective is to see whether a "resident" is running or idle.
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my objective is to see whether a "resident" is running or idle.
Well, the Dock is what most people use, if you can't see the tiny triangles below or beside the running apps, you neede to adjust your screen to fit inside the usable monitor. The only other way I know is the command-tab and the "Force Quit" method, however, neither of those will show the nonrunning but "resident" apps, since they don't display the Dock.
....neither of those will show the nonrunning but "resident" apps, since they don't display the Dock.
Ok. I know which apps "live" in the Dock.
I'd rather push the image to the limits than be able to see the little dots.
I think the Command+Tab thing is better anyway.
Just played with it a bit. It is indeed the case that when the Dock is on the left, the dots are on the left. I can see them. When the Dock is on the right, I can't see the dots, so they must be on the right. With the Dock on the bottom, you get the 3-D "shelf" and the dots are visible. I'm going to try it at the bottom (hidden) for awhile and see if I like it. There's more room for icons along the bottom, so maybe they won't get so tiny as fast.
MamaMoose (hiding behind MaryJo)
I don't think so... got a jpeg of that? 