This appears to be a variation on the boot cache problem with initial upgrade installations of Yosemite.
The difficulty for solving it appears to be knowing precisely when to press and release the Shift ⇧ key to enter Safe boot mode, which remedies the problem.
The procedure is explained in this Apple KBase article.
Shut down the computer by holding in the startup button until it powers down. Start the computer, and immediately upon hearing the startup chime (or when there’s any sign of life such as the sound of a fan, or hard drive, or lighting of the screen), press and hold Shift ⇧ . When the gray apple appears, release Shift ⇧ . After a few moments you will be prompted to log in, and the words “Safe boot” will appear in red letters near the top right corner of the screen.
Be sure to apply the latest OS X update before shutting down or restarting again if your version of OS X is older than 10.10.2. At the time I’m typing this, 10.10.2 is the current version. Having 10.10.2 should prevent the problem from occurring more than once.