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Starting with version 3.0 Thonny+Python bundle is supported on macOS 10.9 Mavericks and later. On older systems you can install Python and Thonny separately.
(Version 2.1.21 can be used on macOS 10.6 and later)
Download latest *.dmg file and open it. You should see a small window with single icon labeled "Thonny" in it.
Drag the icon to your Desktop or Applications folder. This will copy Thonny to your computer.
When copying has finished, you can close the original small window by Ctrl-clicking (or right-clicking) and selecting "Eject":
Now you can (try to) run Thonny by double-clicking its icon on Desktop or Applications folder. Most likely first time your Mac refuses running it, because I haven't paid Apple 99$ per year to prove that I'm a trustworthy developer :p
If this happens, you should Ctrl-click (or right-click) the icon and select "Open" from context menu:
You will get a confirmation dialog where you should click "Open":
You may need to provide password for your (or admin) account. (If you don't have admin rights, then you can use this little trick to make your Mac believe the application didn't come from the internet: https://www.youtube.com/watch?v=I4TrFaYVlB0)
Opening Thonny first time may take several seconds, because Thonny needs to prepare it's files.
Finally you should see Thonny window. Next time you open Thonny, everything should go without any confirmations and much faster.
Allowing apps from anywhere
Starting with macOS Sierra, you may need to do more tinkering: http://osxdaily.com/2016/09/27/allow-apps-from-anywhere-macos-gatekeeper/
Just Ctrl-click (or right-click) Thonny icon on your Desktop or Applications and select "Move to trash".