To install TeXShop, double click the zip file if it did not unzip automatically, and open the TeXShop folder. Drag TeXShop to the Applications folder. Or place it in /Applications/TeX, creating that TeX subfolder is necessary.
For modern TeX distributions, the Path setting preference under the "Engine" tab in TeXShop Preferences should be set to "usr/texbin". TeXShop should do this automatically. If you are using TeXShop with an older TeX distribution or with the Fink or MacPorts distributions, you may have to reconfigure this preference. For instance, if you are using the MacPorts distribution of TeX, this entry should be "/opt/usr/local/bin".
After installing, users upgrading from an earlier version should read "About This Release", the first item in the TeXShop Help menu, to learn how to make minor configuration changes which activate new features.