For Microsoft Windows, MSDN describes the process of doing this here.
I found that part of the implementation is in the registry, but it takes much more than just creating registry keys and values to have your own similar overlay implementation. Dropbox may be actively determining what overlay icons to display while it is continually running in the background, using techniques similar to the Code Project linked by Paul (at least in Windows version of Dropbox).
If you're interested in removing the green check-marks (you only want other status icons like blue sync), you can find solutions for doing it in Windows, Mac and Linux here. It is from here that I partly found out how Dropbox is implementing the overlays.
Best Answer
Backtick (grave accent)
QWERTY
Key that’s been marked with red border. It’s a dead key by default.
This also applies to the Slovak QWERTZ layout.
QWERTZ Germany & Austria
Shift + Key that's been marked with red border
QWERTZ Switzerland
Shift + Key that's been marked with a red border
QWERTZ Czech Republic
Alt Gr + Key that's been marked with red border
QWERTZ Hungary
Alt Gr + Key that's been marked with red border
AZERTY France
Alt Gr + Key that’s been marked with red border
AZERTY Belgium
Alt Gr + Key that’s been marked with red border
Forwardtick (acute accent)
AZERTY Belgium
Alt Gr + Key that’s been marked with red border
Alt codes
There may not be forwardtick keys available for the other keyboards, however an alt code exists (Hold left Alt and type the numbers on your num pad).
The below codes should work on any keyboard.
`´'So if you want an
', you do the following: Hold left Alt, press 3, release 3 but still hold Alt, press 9, release 9, release Alt.Unicode Character 'ACUTE ACCENT'