Under sticky delta, the at-the-money volatility stays at 25% wherever price goes, and each strike's volatility is repriced according to its new distance from the money. The curve translates with the market instead of standing still.
Index markets in trending conditions tend toward sticky delta; single names in quiet conditions tend toward sticky-strike. Real surfaces sit somewhere between and switch regimes, which is one reason volatility books are marked by humans rather than formulas alone.
Example: XYZ at $50 with at-the-money 25% and the $47.50 strike at 28%. XYZ falls to $47.50. Under sticky delta the $47.50 strike is now at the money and prints 25%, while the $45 strike takes over the 28% slot.
Related: sticky-strike, volatility-skew, volatility-surface, vanna