It is not uncommon to see computer terms getting mixed up in everyday discussions. For instance, I regularly hear people referring to a flash drive as simply a "USB," or a full computer system as a "CPU." For some purposes the cross-use of these terms seems to work out, but at other times it can lead to confusion, especially when troubleshooting storage media, when terms like disk (or "disc"), memory, media, filesystem, volume, partition, and mount point can be a headache if used interchangeably.
Ultimately all of these terms refer to some sort of storage device, but when troubleshooting and communicating aspects of storage devices to people it will help to have these terms sorted out. The easiest way to remember drive and drive-handling terminology is to think about three realms: the physical realm, followed by the logical realm, and then the usage realm.
Physical realm (drive, disk)
The first part of your hard drive or any disk drive is the "drive" itself, which like the housing for a floppy drive or CD drive is a device that reads storage media. In the case of a hard disk, the "drive" comprises the metal enclosure and controller boards, heads, and mechanical components that read the disk itself. In the case of flash drives or SSD drives, there is no disk to read but the NAND flash memory medium is still read and written to by various controllers that themselves along with the physical components of the device make up the drive.
The next part of the physical realm of a disk drive is the storage medium, which is the spinning disk (or disc, in the case of CDs, DVDs, and other optical media) or flash memory chips. The medium, regardless of its form, has physical properties that allow it to reliably hold a mechanical or electrical pattern that represents data to the computer.
When looking up hard-drive terminology you may see terms like block (or sector), wedge, track, and cylinder. These describe the layout of how the drive handles its storage medium, since most types of computer storage media are block-level devices that keep items organized in a circular grid for quick indexing and access. These media-specific terms are generally used only when troubleshooting the medium itself (such as when managing "bad blocks").
Basically, in the physical realm, the disk is the thing that holds your data, and the drive is the thing that reads the disk. The physical realm is any component that you can touch with your hands.
Logical realm (partition, filesystem, volume)
Beyond the drive and disk devices are the logical components of a drive, which are how the computer's operating system software handles the device. With a drive connected, the computer has access to a place where it can store files, but first it needs to be able to organize and track the files it stores.
The first logical components of a storage device are the partitions, which are set up in a hidden area of the drive called the partition table. Each partition is a designated section that can be used to hold one volume, and can be used to limit the size of that volume.
Even if a disk or other storage medium is partitioned, the system will still not be able to use it until a filesystem is set up on it. This is done when the partition is formatted, which puts a filesystem structure on it (index files) that makes it possible for files to be tracked on it and thereby turns it into a storage volume. Common filesystem formats are HFS+ (Mac OS Extended), NTFS, and FAT, which are differentiated by the structure and access methods for the index files of that filesystem type (ultimately filesystems are just databases).
Simply put, a partition is a portion of the disk medium that is designated to hold a volume (one volume per partition), and a volume is a partition that has been formatted with a filesystem. The logical realm is any configuration that can be done with Disk Utility.
Usage realm (mount points)
With a storage device installed, partitioned, and formatted with one or more filesystems, the volumes the device now contains will need to be accessible to the operating system. For OS X to access a volume, the volume will need to be given a location through which you can copy files to it. This location is called the mount point for the device and can technically be any logical file path (even a place in your home directory). For the boot volume, the filesystem mount point is always the "root" folder (designated by a forward slash "/"). Folder hierarchies for the filesystem are then built on this, so you have folders like /Applications, /Users, and /System located at the "root" of the drive.
Secondary filesystems such as external disks, non-boot internal hard drives, and network shares are then given mount points within this root filesystem so you can easily access them from the boot drive's filesystem. In OS X, secondary filesystems are mounted in the hidden /Volumes/VOLUMENAME directory of the boot volume (where VOLUMENAME is the name of an individual volume on a disk) and then are displayed by the Finder in various locations such as the Desktop or the Finder sidebar.
The usage realm is any interaction with the disk that is done by applications or services (mounting, file management, ejecting, etc).
Mixing and matching
With all of these terms and organization details, it is easy to mix them up, especially when some of them can be used to refer to other items. For instance, if you referred to the volume "Macintosh HD" on your system as the hard drive, then you would be correct; however, if you remove the hard drive from your system and call it a "volume" then that would not necessarily be correct, since the drive can contain multiple volumes.
One approach to keeping these terms organized is to see them hierarchically in the following manner: Drive --> Disk/Media --> Partition --> Filesystem/Volume --> Mount Point, where if you start with any point on the list then that means all the items before it in the list must be present in some form, and that items after it in the list are not necessarily present. For example, if you have a disk in your system then you definitely should have a drive to read it; however, it does not necessarily mean you have the disk partitioned and set up as a volume. If you have a filesystem or volume then that does mean that there is a partition for it, storage medium for it to be on, and a disk to read that medium; but it does not mean the filesystem is mounted and has a mount point.
Overall it is very easy to overlap these terms, and while often it is easy to figure out what people are referring to, at other times (and especially when there are complicated multidisk setups on a system) it may be easy to get confused. Nevertheless, with a little clarification you should be able to get things straightened out.