A falling knife is a slang phrase for a security or industry in which the current price or value has dropped significantly in a short period of time. A falling knife security can rebound, or it can lose all of its value, such as in the case of company bankruptcy where equity shares become worthless.
A falling knife situation can occur because of actual business results (such as a big drop in net earnings) or because of increasingly negative investor sentiment.
As the phrase suggests, buying into a market with a lot of downward momentum can be quite dangerous. If timed perfectly, a buy at the bottom of a long downtrend can be rewarding - both financially and emotionally - but the risks run extremely high. This term implies that the investment will never be a good one again. Examples of stocks that have plummeted are plentiful; a widely-held stock can drop precipitously as the equity ownership is reduced to nothing.