If you have a phone or laptop giving you trouble and need to troubleshoot the device, a serial number will give the person assisting you a quick way of looking up your device. Serial numbers are supposed to be unique, but they are usually assigned in such a way that they can be tied to a specific lot or batch of devices. A tablet's SN can tell the company which model of tablet you're using, if you yourself are unable to answer that question.
Similarly to how the S/N can be tied to a specific batch, it can also be checked against the vendor's database to determine the time of sale and hence determine warranty status. Even if you don't have the receipt for when you purchased the device, some companies can tell you the item's warranty status just by checking the serial number.
Finally and most importantly, the serial number establishes ownership. If your device is stolen and shows up on eBay or Craigslist, simply telling the authorities, "That looks like my camera that was stolen!" will likely fail to spur them to action. Presenting the cops with photographic evidence of the stolen goods for sale and proof of ownership, however, might turn the tide in your favor.
Take a look at all of your electronics: your laptops, your phones, your tablets, your gaming consoles, your cameras, your mp3 players (if you still have one of those), even your OS and software license keys. If they are of value to you, make a list of them.
Make a file in Notepad/Excel/Evernote/Dropbox/wherever where you write down each of the following for all of your important and/or expensive devices:
|Canon||EOS Rebel T6||XXXX-XXXX-XXXX-XXXX|
|Laptop||Toshiba Satellite S55-C5274||YYYY-YYYY-YYYY-YYYY|
|iPhone||4S - Unlocked||A52137189|
If your device is stolen, the only way to definitively prove that the item for sale on Craigslist is yours is if you can present a picture of it and its serial number. Take pictures of the device itself (so that you do not attempt to claim someone has stolen your laptop using the serial number of a toaster), and take a good-quality photograph of the serial number on the back of the device. Keep these photos with the other documentation on the device, and keep them both somewhere safe. It just might make the difference between the police helping you retrieve your camera or not.