What does a "sell-off" mean if every stock transaction is balanced? If every stock transaction is "balanced" (1 sell = 1 buy), then what does a "sell-off" actually mean and how does it move the price of a stock? 

Ex. If millions of people sell a stock, there's necessarily an equal millions of buys of that stock.
The Cambridge English dictionary has a definition here: link to definition

I will paraphrase their definition: a sale of equity shares that causes its price to fall.

As the question notes, that appears to make little sense, if for every buyer, there is a seller. If stock prices were negotiated by individual verbal contracts, the implication was that the seller was more morivated to sell than the buyer to buy, and so the price had to be adjusted lower.

However, the common usage of “sell off” in the financial press is different: it is a synonym for a stock price (or stock index) experiencing a price fall on a day versus the previous day. The implication is that all the sellers were motivated than buyers, and so the price fell, like in the verbal contract.

However, if the observed price on the exchange is lower, we only know two things:


For every buyer, there is a seller.
The price of the transactions on the current day is lower than the previous day.


There’s any number of ways the price can end up lower, and short of questioning everyone involved in trading that day, we have no idea what really happened.

I will now give an example how intuition about supply and demand can be misleading. 

Imagine that Acme Inc. traded yesterday at \$10. The next day, market makers set the price around that price, with a bid-offer spread. There is no trading.

Then imagine that at 10 am, Acme announces a successful lawsuit against it by the plaintiff Wile E. Coyote. The market makers are shocked, and drop the mid of the bid/offer prices to \$5, and still there is no trading.

Meanwhile, retail stock traders who did not know about the lawsuit saw the new price, and rushed to buy. 100 customers buy 100 shares each for the market makers, who are happy to sell out of inventory.

From the persepctive of an outside observer, we saw 100 retail traders rush to buy shares - and the price fell by half.