Modern computers have dedicated RAM for running programs, and large volume storage to record program outputs and to store data when not in use. Large volume storage trades access speed for capacity; and it comes in three broad categories; internal to the computer, external to the computer and storage that is accessed via a network.
Internal storage is a device housed inside the computer. It comes in two broad types, as of 2014; spinning disk hard disk drives and solid state hard drives. Spinning disk drives have platters of magnetic material that get written and rewritten by a drive head. Solid state disks use flash memory to store data. Solid state disk drives are much faster than spinning platters, and can fit unusual case shapes, like the MacBook Air or Ultrabook, for example. Solid state drives are much more expensive for a given capacity than a conventional spinning disk drive.
External storage devices include removable media like optical disks and SD Cards and USB-connected flash drives (called "thumb drives" because of their size) and externally connected hard drives. External storage is used to back up computer data, move files between a computer and a different machine without a network, and make copies of something for other users. As of mid-2014, optical disks are on the way out in consumer devices, external hard drives are used for backing up systems, and thumb drives are used for most conventional file transfers.
Networked storage devices included shared file servers, network accessible storage (NAS) devices and cloud services. All of these take the idea of an external hard drive and change the connection method from USB to a full network connection. You add file servers to your local domain and have total control over who accesses which files on your server. You can plug an NAS device to your router and every computer on your local network can store files on it for a more informal file sharing method. You can also use services like DropBox, Box and SkyDrive to put files up on another vendor's servers and access them through the Internet.