How can a secret be split so that no single person holds it?

You can split a secret into pieces, called shares, so that no single person holds anything intelligible about it, yet enough people putting their shares together can rebuild it. This is secret sharing. In a well-made scheme it is "all or nothing", where "all" means the necessary number of shares: with fewer, you know no more than someone holding nothing at all.

The reason to bother is a dilemma that anyone guarding a precious key faces. Keep one copy in one place and it is as secret as it can be, but lose it and the secret is gone. Keep many copies in many places and you are safer against loss, but every copy is one more chance for the secret to fall into the wrong hands. Splitting the secret escapes the trap. It even lets you trust a group in aggregate: the secret stays safe as long as the number of traitors never reaches the number needed to reconstruct it.

Picture a company protecting its secret formula. The president should be able to open it whenever needed, but in an emergency any 3 of the 12 board members could unlock it together. A threshold scheme gives exactly that: any group of t or more of the n players can rebuild the secret, and no group of fewer than t can.

Cutting a secret up carelessly does not work. Split the word "password" into four two-letter pieces, and the first piece alone leaves only six letters to guess, about 308 million combinations instead of the 208 billion facing someone with no piece at all. Each piece leaks, so that scheme is not secure. Proper secret sharing was invented independently by Adi Shamir and George Blakley in 1979.