To format a drive essentially means to wipe that drive clean of all data (generally because you want to use it for another purpose). Mac OS X has a great utility built-in that allows you to format drives. It's called Disk Utility (Applications > Utilities). To format a drive on OS X, you'll first want to launch Disk Utility. Once Disk Utility is open, you will see a list of drives available on the left side of the screen. Select the drive you want to format (be careful not to select a partition, which will be indented in the list). Click on the "Erase" tab, choose the type of format to use, then Erase... It will look something like this:
A word about formats: if you plan on using the drive on both Mac and Windows PCs, then you will want to choose the "MS-DOS (FAT)" format. Although it's not a very efficient format to use, it is readable and writable on both OSes. If you plan on using the drive on a Mac only, you definitely want to choose the "Mac OS Extended (Journaled)" format. It's a much more efficient format, but isn't accessible on Windows.
For most, the process of formatting a hard drive means the data is gone at this point. Unfortunately, that isn't technically true with a quick format like this. Formatting a hard drive (really formatting a hard drive) takes time. The average user is more concerned with convenience, and less about security. As long as we see a message at the end indicating that the drive has been formatted, we're good. Right? Well, with the right tools, that formatted drive is still completely readable by users with not-so-good intentions.
If you're looking for more than a quick format (in other words, you really want your data gone), there are options for that as well, and it's all built-in to Disk Utility. Before clicking Erase... you'll notice a button next to that called "Security Options...". Clicking on this will expose several options for how your drive will be formatted. Below are the four (4) formatting options to choose from:
As you can imagine, the more secure the format, the longer it will take. As I said before, really formatting a drive takes time, but can be worth it (especially when we're talking about sensitive data). As a note, the "quick format" that I referenced earlier is the default, which is also the "fastest" selection on the sliders shown above. If you really, really, really want your data gone, slide the slider all the way to the right and erase. The only thing more secure than this is an industrial shredder. I've seen them in action. Trust me, the data is really gone then.