Normally when you boot OS X, the system displays a gray screen with an Apple symbol and a revolving progress indicator, and then may display a blue screen before loading the log-in window and default background. These screens are just coverings for the activity that is happening during boot, and make OS X more visually appealing.
While most people may not need to see what is happening every time the system starts up, if your system takes a while to load or has other problems at boot such as crashing or hanging, then seeing where this is happening can help. At other times it may be useful to check that certain items have loaded, especially if you have a custom or specialized hardware setup.
Enabling verbose mode
As part of the OS X boot process, boot argument variables are passed to the kernel as it loads. These variables are either stored in the system's PRAM or temporarily passed to the system via key presses during boot. The system will look for key presses and will also pass all variables stored in the PRAM to the kernel during boot; since usually these variables and key presses are missing, the system then boots normally without any modifications.
The quickest and easiest way to pass the "-v" (verbose) boot argument to the kernel is to hold Command-V immediately after you hear the system's boot chimes. This will send the -v argument to the OS X kernel that will tell it to boot into verbose mode.
While this is a quick option and likely the easiest for most purposes, you will have to hold the keys down each time the system boots if you want to get into verbose mode. This can be a bit cumbersome, so you may prefer to just store the -v argument in the system's PRAM so it is loaded every time the system boots. To do this, launch the Terminal application and run the following command (Note: NVRAM is the same as PRAM):
sudo nvram boot-args="-v"
With this variable set, any time you boot the system you should see the output of the system loading each component.
Stuck in verbose mode?
Sometimes systems may be stuck in verbose mode (or another boot mode). If this happens it is likely because of faults with either of the two methods of sending boot arguments to the kernel.
The first and more likely possibility is that the system's PRAM is corrupted and is not passing empty boot arguments to the kernel. Therefore, try a quick PRAM reset by restarting and holding the Option-Command-P-R keys. With these keys held, allow the system to cycle through a couple of resets and then release the keys and allow the system to boot normally. At this point the system should not boot into verbose mode or any other mode.
In addition to resetting the PRAM, you can try manually setting and clearing the "boot-args" variable, by running the following two commands sequentially in the Terminal:
sudo nvram boot-args="-v"
sudo nvram boot-args=""
A second possibility is that there may be a problem with the keyboard attached to the system. If there is a fault and the system is detecting a key press during boot, then this could result in the system loading with an undesired boot argument. The best way to test this is to either unplug the keyboard when booting or to try a new keyboard with the system.
Does verbose mode change anything?
Being an alternate view of the system during bootup, some people may think that verbose mode is another operating mode like safe mode that may limit the system or somehow cause it to operate in a different manner. This is not the case, and the system should load the same as it always does, only now it shows you what it is loading instead of veiling it behind a blue or gray screen.