A hard drive needs to be initialized by the operating system before it can be used. The process of initializing the hard drive involves setting the partition style; this will define how the hard drive will store the partition information so that the operating system knows which sectors belong to each partition, and which partition is bootable. The options are MBR (Master Boot Record), and GPT (GUID Partition Table), with GPT being used more commonly with newer, larger capacity hard drives.
A hard drive needs to be partitioned so that the operating system knows how the data on the hard drive will be arranged. It is common to create one large partition on a hard drive for all of the data (often identified as C:). You can create multiple partitions on a hard drive, and each will be assigned a drive letter in Windows or a name in Mac OS.
Formatting a hard drive is necessary to apply a file system. A file system is used to control how data is stored and retrieved. In Windows, the most commonly used file system is NTFS, and for Mac OS it is Mac OS Extended (also referred to as HFS Plus).
Note: Formatting a hard drive will erase all data on the hard drive.  Make sure you have any data backed up before continuing.