With some external hard drives being shipped in a FAT 32 (PC) format, it is recommended that they be formatted for Macintosh's native format (Mac OS Extended or Mac OS Extended (Journaled)) when backing up Macintosh volumes.
When you enter Disk Utility to initialize or erase the new volume, it does not have the option to format as "Mac OS Extended".

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To format the external hard drive for "Mac OS Extended", first select the main volume of the external hard drive in the left column of Disk Utility.
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Then click on the "Partition" tab.

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At the bottom of the screen, click on the "Options" button.

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Change the selected option to "Apple Partition Scheme" if connected to a PowerPC or "GUID Partition Table" if connected to a Intel-based Mac then click "OK".

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Now, click on the "Erase" tab and select a Macintosh based filesystem.

If setting up a "Bootable Backup", then stay in the Partition tab and create the partitions (typically, one for the bootable backup and one for storage). Once, finished, click "Erase" to proceed.
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Once the filesystem is selected, click the "Erase" button. A WARNING will come up saying that this will erase all data on the drive---click Erase to proceed. The formatted volume will mount on the desktop.

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Close Disk Utility.
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Verify that the formatted volume is Mac OS Extended by clicking on the volume (once) and hitting the Command {APPLE) + I keys, OR go up to File and then click Get Info. The "Format" will show the filesystem.

If you are setting up a Bootable Backup, you will need to uncheck the "Ignore ownership on this volume" box.