A short definition
EXIF stands for Exchangeable Image File Format. It is a standard for storing structured information inside an image file, written by the camera at the moment of capture and carried along by the file forever after. It was designed so photographers could sort their own libraries. It was not designed with sharing in mind.
What is usually in there
Camera make and model, lens model, exposure settings, orientation, the date and time to the second, and the software version that processed it. On phones, almost always GPS coordinates as well. Professional bodies add a serial number. Some cameras record the owner's name, because the owner typed it in once during setup and forgot.
Why the serial number matters more than it sounds
A camera body serial number is a persistent identifier. Every photo that body ever took carries the same one, which means a set of otherwise unconnected images can be tied together and traced back to a purchase record. Anonymous photographs are rarely as anonymous as intended.
EXIF is not the only layer
Editors add their own. Adobe writes XMP, a second metadata block in XML. Photoshop writes an IPTC record. Each one can hold a name, a copyright line, a city, and a document tracking ID that survives across saves and edits.