Standard deviation punishes a plus 9% month exactly as hard as a minus 9% month. Downside deviation counts only the periods below a minimum acceptable return - usually zero or the risk-free rate - and treats everything above it as zero.
The calculation: for each period take min(return - threshold, 0), square it, average over all periods including the good ones, take the root. Note the denominator uses all periods, not just the negative ones; using only negative periods is a common error that inflates the result for strategies with few losses.
It is the denominator of the sortino-ratio, and it particularly changes the ranking of positively skewed strategies - trend following looks far better on downside deviation than on total volatility, because most of its variance is upside. It does not fix the tail problem: a strategy with rare catastrophic losses still shows a small downside deviation until the catastrophe arrives.
Related: sortino-ratio, return-skew, sharpe-ratio, ulcer-index