"Desktop" drives are all the "3½" Form Factor" units that are the commonly-seen hard drives. They are about 3½" wide, and intended for mounting internally in a computer case. "Laptop" drives are smaller physically - typically about 2½" wide - to fit inside the small cases of those machines. Because of their physical size they usually have smaller storage capacities and are more expensive. You're less likely to see these on a retail shelf, but you certainly can buy them to add to a machine.
External drives are just normal internal drives like the above, but mounted inside its own case so it is NOT mounted inside your computer. It comes with a data cable to connect to the computer, and most also have their own power supply arrangement, too - either a cord, or maybe a "wall wart" and cord. Some externals, particularly smaller portables intended for use as laptop accessories, can get all their power from the computer host via their USB cable and do NOT have their own separate power cable. "Notebook Drive" typically is used to describe a small portable external drive intended as an accessory for a laptop or notebook computer.
You can buy external drive units that are complete and ready to go, with the drive unit already mounted inside the case, etc. Alternatively you can buy separately your own case and HDD unit and assemble them into the finished product.
The data cable connection between a computer and the external unit could be any of: USB (usually USB2, the newest version), eSATA, IEEE 1394a (aka Firewire 400), or IEEE 1394b (aka Firewire 800); the Firewire 800 is less common. USB2 is almost universal, but is the slowest of these options. To use any of them you need matching ports on your computer and your external drive and the right cable. Many externals come with 2 or 3 of these, and you can use whichever is suitable at any time - just use only one connection system at a time.
If you have an external drive the way you connect it to your computer, and the type of computer you use, is determined entirely by this data interface matter. A so-called "Notebook Drive" can be connected to ANY computer as long as they each have matching ports for connection. A larger External Drive commonly used with a desktop machine also can be transported and connected to any laptop with a matching port - it just takes up more space in your suitcase than a smaller "Notebook Drive".
All hard drives, internal or external, need a little preparation before they can be used by your computer's Operating System. First you must create on it one or more Partitions (chunks of space to be used as one logical "drive"), then you must Format each such "drive". You have the option of making the entire HDD space into one large drive, or of making more than one on a physical device. Often these two steps are considered one operation called "Initializing" the drive. Windows has tools for this built into it in Disk Management. Hard drive manufacturers also will give you free software utilities to make this process easy. Until you do these things, no OS will be able to "see" the drive and use it for files, even though it is detected as a valid hardware device in the system.