Drag it Away
The easiest is to simply drag the icon upwards out of the Dock.  As you begin this process you will notice a small puff of smoke on the icon.  Once you have moved it completely out of the Dock release the mouse button to see  the icon vanish with an animated puff of smoke.  Mac OS X has imposed restrictions as it will not allow you to remove the Finder or the Trash from the Dock.

Removing a running Application
There is however a unique behavior when you attempt to remove an active application (indicated by a black triangle) from the Dock.  You can complete the procedure to remove it but it will remain in the Dock.  When you quit the application the icon will disappear.

Use Dock Menus
You can remove an icon from the Dock two different way:

1. Place your mouse over an icon in the Dock then click and hold the mouse button
2. Press Control and simultaneously click on the Dock icon

You'll soon see a subdivided menu appear above the icon.   Note that if you have repositioned the Dock to the left or right, the menu will appear to the right or left respectively. The topmost option is "Remove from Dock" and if selected the item will be removed.  The behavior is identical on both sides of the Dock divider.

Updates

• February 27, 2009 - content revision, updated images