The “Fear Index” or “Fear Gauge” are nick names for the VIX Index.
Why is it called the Fear Index?
Usually the VIX Index rises when the stock market falls, and conversely, the VIX Index will fall when the stock market rises (= less fear). Typically, there is an inverse correlation with the S&P 500 Index.
In other words, when there is fear and uncertainty in the U.S. stock market, which is reflected in falling stock prices, the VIX is usually high and vice versa.
The higher the fear, the higher the VIX Index. Therefore, the nick name “Fear Index”.