How do I install and play Dwarf Fortress on mac?

From the README.osx:

DF is not currently bundled as a proper Mac application. You'll need to use the "df" script in the folder. Double-clicking on it should work. You can also try running it from the Terminal using "sh df" from the Dwarf Fortress directory.

If you run Terminal, you'll need to do so in the df_osx folder. I'm sure there is a mouse–cursor menu item that says something like “Open a Terminal window here ”.

That should do it.

To be blunt, I would recommend that you hold off on too much more actual installing until you learn a little more about what you are doing.
Really, that makes sense, and isn't disparaging in any way: Trial–and–error is a good way to learn, but installing new executable programs to your computer isn't the best arena for that.

1. Download the Mac version from http://www.bay12games.com/dwarves/older_versions.html ( current version: http://www.bay12games.com/dwarves/df_43_05_osx.tar.bz2 )
2. save somewhere you can find it, like a folder for ~/Games
3. double click the df_43_05_osx.tar.bz2 file (or whichever version you downloaded) to extract.
4. you should now have a folder/directory named df_osx inside your Games folder. open terminal.
5. type cd ~/Games/df_osx and hit return. this changes the terminal to be inside the dwarf fortress directory. ( see https://stackoverflow.com/questions/9547730/how-to-navigate-to-to-different-directories-in-the-terminal-mac )
6. now that you are in the dwarf fortress directory in terminal type ./df and press enter. this runs the file named 'df' inside the current directory './'.
7. if this doesn't work it could be you need the 32bit instead of the 64bit (or vice versa). download the correct one and try again. you should now have a !!FUN!! intro playing.

!!Enjoy!!

I highly recommend The Lazy Newb Pack for Mac. It includes DFHack (the premier mod and one almost all other mods rely upon), alternative graphics and tile sets (which are much nicer than the default rectangular ASCII 'graphics'), and a nice launcher app too.