What the other posters have described, is accurate, but, somewhat, just the tip of the iceberg.
A running configuration is saved as a file. On a Cisco devices, a file might be written to many different places, many supported by the copy command. Besides being able to copy to network locations using a protocol like tftp or FTP, some platforms support other protocols too.
Also besides copying a file to a network location, you can copy it to different local (writeable) memory devices, some removable (latter possibly using USB sticks, SD-card or CF-card).
As Balaji also mention, you might just display your configuration to your interactive session. Then you might screen copy it or some interactive sessions support "logging", i.e. save to a local file everything displayed.
The prior assume an, upon demand/need, manual intervention. But, as Balaji also mentions, auto backup might be done with some form of scripting, either on or off the device. Also, various network management platforms will either save copies of devices configs, on a reoccurring basis, and/or, when management device "notices" device config has been changed (the latter is especially nice, also somethings a history of configs is saved so you can easily compared what's changed between saved configs).
