Higher order Greeks are option Greeks other than first order Greeks (first order Greeks include delta, theta, vega, and rho).
They measure sensitivity of first order Greeks to factors like underlying price, volatility, time, or interest rate, in a similar way that first order Greeks measure sensitivity of option price to these factors. Mathematically, higher order Greeks are higher derivatives of option price with respect to these factors.
The best known higher order Greek is gamma, which measures sensitivity of delta to small changes in underlying price (so we can say gamma is "delta of delta").
Higher order Greeks include second and third order Greeks.
List of Higher Order Greeks
Charm = Also delta decay or DdeltaDtime. Sensitivity of option price to small changes in underlying price and passage of time, sensitivity of theta to small changes in underlying price, or sensitivity of delta to passage of time.
Color = Also gamma decay or DgammaDtime. Sensitivity of gamma to passage of time (small changes in time to expiration).
Gamma = Sensitivity of option delta to small changes in underlying price.
Speed = Also DgammaDspot. Sensitivity of gamma to small changes in underlying price.
Ultima = Also DvommaDvol. Sensitivity of vomma to small changes in volatility.
Vanna = Also DvegaDspot or DdeltaDvol. Sensitivity of option price to small changes in underlying price and volatility, sensitivity of vega to small changes in underlying price, or sensitivity of delta to small changes in volatility.
Vera = Also rhova. Sensitivity of option price to small changes in volatility and interest rates, sensitivity of rho to small changes in volatility, or sensitivity of vega to small changes in interest rates.
Veta = Also vega decay or DvegaDtime. Sensitivity of option price to small changes in volatility and passage of time, sensitivity of vega to passage of time, or sensitivity of theta to small changes in volatility.
Vomma = Also vega convexity, volga, or DvegaDvol. Sensitivity of vega to small changes in volatility.
Zomma = Also DgammaDvol. Sensitivity of gamma to small changes in volatility.