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Like Windows, but unlike OS-X and Linux, the Amiga has a multi-rooted filesystem. That is, you don't have a single “/” root directory that all other filesystems are found under.
On the Amiga, all of the above have alphanumeric names. When you see them used, they usually end with a “:”.
Devices includes things like SER: (serial port), CON: (console/shell) or DF0: (first floppy disk), DH0: (first hard disk). The equivalent on Linux are the files under /dev/.
Volumes are actual filesystems mounted with a name. E.g. commonly System:, Work: etc., that refer to a filesystem on a partition on one of the disk devices.
Assigns are best described as temporary aliases. E.g. LIBS: (equivalent to /lib, /usr/lib etc. on Linux), DEVS: (equivalent to device driver modules under /lib/modules/ on Linux). You can create your own assigns.