How can I rename many files at once?

In versions of OS X before Yosemite, renaming a group of files at once either meant third-party software or rolling your own rename script using something like Automator or AppleScript. Now, though, you can just select a group of files and then select Rename either from the right-click contextual menu or from the drop-down button marked with a cog icon in Finder windows.

When you do, you get the option of adding text, replacing text, or applying a format such as a name and an automatically incrementing counter.