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Normal Format -
When you choose to run a regular format on a volume, files are removed from the volume that you are formatting and the hard disk is scanned and cleared of bad sectors. The scan for bad sectors is responsible for the majority of the time that it takes to format a volume.

Quick Format -
If you choose the Quick format option, format removes files from the partition, but does not scan the disk for bad sectors. Only use this option if your hard disk has been previously formatted and you are sure that your hard disk is not damaged.

Note: I use the windows xp install format as an example but it is the same concept when formatting a flash drive, external, etc..