Three things are being declared here: an anonymous enumerated type is declared, ShapeType is being declared a typedef for that anonymous enumeration, and the three names kCircle, kRectangle, and kOblateSpheroid are being declared as integral constants.
Let's break that down. In the simplest case, an enumeration can be declared as
enum tagname { ... };
This declares an enumeration with the tag tagname. In C and Objective-C (but not C++), any references to this must be preceded with the enum keyword. For example:
enum tagname x; // declare x of type 'enum tagname'
tagname x; // ERROR in C/Objective-C, OK in C++
In order to avoid having to use the enum keyword everywhere, a typedef can be created:
enum tagname { ... };
typedef enum tagname tagname; // declare 'tagname' as a typedef for 'enum tagname'
This can be simplified into one line:
typedef enum tagname { ... } tagname; // declare both 'enum tagname' and 'tagname'
And finally, if we don't need to be able to use enum tagname with the enum keyword, we can make the enum anonymous and only declare it with the typedef name:
typedef enum { ... } tagname;
Now, in this case, we're declaring ShapeType to be a typedef'ed name of an anonymous enumeration. ShapeType is really just an integral type, and should only be used to declare variables which hold one of the values listed in the declaration (that is, one of kCircle, kRectangle, and kOblateSpheroid). You can assign a ShapeType variable another value by casting, though, so you have to be careful when reading enum values.
Finally, kCircle, kRectangle, and kOblateSpheroid are declared as integral constants in the global namespace. Since no specific values were specified, they get assigned to consecutive integers starting with 0, so kCircle is 0, kRectangle is 1, and kOblateSpheroid is 2.