If you have less than 16 MB video RAM. Blender only runs OK when you set your display on "thousands of colors" (16 bits). Systems with fewer than 8 MB won't even run Blender, for that we have provided a warning message at startup per Blender version 2.34.
If other applications use OpenGL, or when you have a very large deskop resolution, or use 2 monitors, you might run into troubles as well.
You can do tests with a small Blender window size though. Do this by starting Blender from a Terminal. Type here the full path to the Blender.app and add to that;
<path_to_blender.app>/Contents/MacOS/blender -p 100 100 700 500
This opens a small 700x500 pixel Blender window, offset 100 pixels from the bottom-left corner.
OSX Aqua uses OpenGL extensively for drawing, so it might happen your 3d card doesn't have sufficient memory to draw accellerated with a larger window size. A solution that works well, is switching to less screen colors: set the menu in "System Preferences -> Displays -> colors" at "thousands of colors".
Another common issue is with using the Dock on top of the Blender window. This kind of transparent blending takes a lot of 3d card drawing powers, and can better be avoided by setting the Dock to become auto-hide.
Unfortunately Apple has released a series of cheap Macs and Macbooks with only bare OpenGL support. Especially Mac Minis and Macbooks with intel graphics card lack the support to use a 3D application like Blender.
Blender is programmed to be used with a 3 button mouse. When you are using the standard 1 button Apple mouse this functionality can be accessed by keeping keyboard keys pressed while pressing the mouse button.
If you have a three button mouse, or a mousewheel as 3rd button, you might have configured the 3rd button to another action like "double click". Blender can only detect the middle-mouse button if you configure your mouse device correctly.
Check the "Other" category in "System Preferences". For a Logitech mouse, you can assign it "Advanced Click" and with Device Button Number "3".
Blender runs at many different platforms, and all versions are utilizing OpenGL to draw the entire interface. This keeps the program portable, and makes it easier to maintain.
This is the reason why the look and feel of Blender is quite different from standard Aqua applications. Attempts to integrate Blender better with the OSX desktop are being in process. However, for the short term we don't expect a full Aqua version of Blender.