How to find your Switch's serial number
It's a sticker on the bottom of the console, near the USB-C port.
If it's still boxed
You don't need to open the box. Nintendo prints the serial number in two places on the packaging: through a small cut-out window, and on the barcode sticker on the outside.

If it's already unboxed
Turn the console over. Near the USB-C charging port, on the bottom edge, there's a white sticker with the serial number printed on it — it starts with an X.

This applies to the original Switch, Switch Lite, and Switch OLED alike — same sticker, same spot.
Reading the format
A full serial is a 4-character prefix followed by more digits, e.g. XAW10012345678. The prefix tells us the model and manufacturing batch; the digits that follow place it within that batch. The checker only needs the prefix plus the first 6 digits — paste the whole thing in and it'll take care of the rest.
Have it? Head back to the checker and paste it in.