1. A post that holds a reel of magnetic tape or CDs. In the picture to the right, is an example of a blank CD spindle. When purchasing blank discs they'll often be sold in a spindle like the one shown here.
2. In the case of a hard drive, as seen in the below picture, the spindle is what holds a hard drives platters in place. With a traditional hard drive there is the need to have multiple platters. The spindle holds these platters in a fixed position with enough space for the read/write arms to get to the data on the disks.