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In brief, Cisco devices often support 16 privilege levels. Commands can be tied to specific levels. By default, Cisco devices allow just a couple of commands at level 0, allow many to most "show" commands at level 1 (also commonly called user mode) and allow every command at level 15 (also commonly called privileged mode).
As often only levels 1 and 15 are used, Cisco devices also change the command prompt's last character to indicate which level you're in. "#" indicates you're in privileged mode, ">" indicates you're in user mode. (Actually, levels 0 and 1 appear to display ">" and levels 2..15 appear to display "#".)
Configuration mode is a different command input mode. This mode allows you to change the device's running configuration. Getting into this mode requires your privilege mode level allow it, and when entered, the tail of the prompt string, initially, starts with "(config)#". There are submodes to configuration mode, for example, if you're changing an interface's configuration, the tail of the prompt string has "(config-if)#".
