It's not unlike fractional reserve banking.
Imagine three participants: Alice, Bob, and Charlie. Alice owns a share of Foo Corp, the only such share that exists. Bob decides to short Foo Corp, and borrows a share from Alice's broker (Alice's broker gets paid for this).
At that moment, there are two shares that exist. If Alice logs into her brokerage account, she sees a share of Foo Corp, but it's really an IOU for a share of Foo Corp. If Bob logs into his account, he also sees a share of Foo Corp (that he borrowed from Alice). At that point, there are two "shares" of Foo Corp that exist.
To make money on the short, Bob has to sell the share and buy it later, so he puts it in the market and sells it to Charlie. At this point, there are two shares: a share owned by Charlie and an IOU to Alice from Bob.
Bob could be very confident about shorting Foo Corp, so he could also borrow a share from Charlie's broker, at which point there would be three shares in existence: an IOU to Alice, an IOU to Charlie, and the sole share that exists. Each of these participants, if they log into their brokerage account, do see a share.
These phantom shares can't exist forever though: Bob is paying both Alice's and Charlie's brokers. Keep in mind that either Alice and Charlie can decide to sell their share, which Bob could then buy, at which point he could cover the remaining IOU, and everything would settle.
But then, what if both Alice and Charlie just bought the share and weren't planning to sell it? Bob would be in a position where he owes two shares but there are zero to buy on the market. Normally, that doesn't happen because shares are traded often enough, but in certain scenarios (short squeeze, collusion between participants) there might be issues. At that point, Bob can decide to keep on borrowing the shares until one sells, or maybe he runs out of cash to borrow shares and closes his account.
That's when things get a bit dicey. At this point, there are two "shares", one actual share and one IOU for a share. What happens is that Bob's broker is in a fail to deliver position. They're supposed to deliver a share of Foo Corp to close the IOU but there's no such share to deliver.