Sensitivities of sensitivities
Delta, gamma and vega change as price, volatility and time change. Higher-order Greeks put a number on those changes. Vanna is the change in delta when volatility moves. Charm is the change in delta as time passes. Vomma is the change in vega when volatility moves.
Speed is the change in gamma when price moves. Zomma is the change in gamma when volatility moves. Color is the change in gamma as time passes. Veta is the change in vega as time passes, and ultima is the change in vomma when volatility moves.
An example
Take an at-the-money option a few days from expiry. Its gamma is high and its color is large: each day that passes raises its gamma further, if the price stays near the strike. That is why the exposure map near expiry can change shape from one day to the next even when open interest barely moves.
Two related measures
Dual delta is the change in the option's price when the strike changes, rather than the stock price. For a call, its size equals a discounted probability of finishing in the money implied by option prices. That probability comes from prices, which build in what investors pay for risk, so it is not a forecast.
Lambda is leverage: the percentage change in the option's price for a 1% move in the stock. Caveat: lambda becomes unstable for very cheap options, and every time-based Greek depends on a convention. Measured in calendar time and in time remaining, the same Greek has opposite signs, so always check which one a number uses.