signature
A signature on a PDF is a seal inside the document that proves who approved it, and that not one byte has changed since.
What the word means
It is not a picture of a handwritten name. A picture proves nothing: anybody can paste one in.
A signature is a number worked out from the whole file with a private key that one person or one company holds. Anybody can check it with the matching public key, and nobody can produce it without the private one. Change a single character in the document and the check fails.
What it does on a real document
The seal covers the whole file except the small hole its own value sits in. A reader opening the document reads everything on both sides of that hole, works the number out again, and compares the two.
This is also why a sealed document can be added to without breaking the seal: what is added is written after the sealed part, and the reader shows the document as it was approved and says what came afterwards. The engine leaves the hole and writes the file around it; your key never leaves your own machine.
Where it comes from, and why
The idea is older than the format. A way of proving who wrote a message, without the two sides ever having met and without a shared secret, was published in the 1970s, and everything since rests on it.
PDF gained a place to put such a seal early on. Europe has since written the rules for what one has to carry to stand up in law, a set of rules known as PAdES. The seal alone is only half of it: the other half is the certificate saying that the key really belongs to whoever claims it.
See it on a finished document
The rules behind this word, and the texts they are read from