The error literally means what it says. That the activity that runs when you press your home button (or return to the home screen) is not responding. If you click "Wait" a few times, and it still comes back with that message and isn't responding you might have a problem.
When you go back to the home screen (desktop), your phone has to do a bunch of things. Processes don't remain running in the same way as they do on computers or other phones. Instead, their "state" is saved. When you go back to the application, the state is loaded and it all looks like it was before.
This saving of the application's state takes some time. Especially if it's something that uses a lot of memory like the browser, or perhaps a game.
Meanwhile, there are other processes running. Some of them it has to deal with immediately. These would include managing connections, dealing with any incoming calls/messages, acknowledging signals from the network, if the GPS is running it'll be calculating stuff related to that, etc.
Also, the home activity itself is trying to do a bunch of things... clear your screen, redraw the desktop image, draw the icons on top of it, restore the desktop's state (e.g. if the menu was open, etc). A previous poster mentioned he was trying to scroll the desktop at the same time too - all these things use up the CPU power of your phone.
While all of that is going on the operating system is timing how long all these activities are taking. If they take over a set threshold (usually a few seconds at most) the operating system gives you the option of closing or waiting for the activity to complete. This is what you're seeing.
Hence, if clicking "Wait" works, even after several waits, it just means your phone was busy doing stuff it has to do.
If it clicking Wait several times doesn't stop the error displaying, then indeed the activity process might have hung, and in the case of the activity you mention, closing it is fine as the operating system will automatically restart it as needed.
Why is your phone getting slow? Well, activities running in the background from various applications you install can add up. If the phone is low on memory the operating system tries to close as many activities as it can to free up memory... this might mean a whole bunch of activities are trying to save their state at the same time.