The Greeks are the mathematical derivatives in financial derivatives^{♣} and options.
They are called Greeks because they are all known by letters of the Greek alphabet. Traders and risk managers
are interested in the Greeks because they tell them how much or how often they have to hedge. The most
common Greeks are:

Delta: the rate of change of the derivative price with respect to the underlying asset price (sometimes known as the hedge ratio).

Gamma: the rate of change of the Delta with respect to the underlying asset price.

Theta: the rate of change of the derivative price with time.

Vega (or Lambda, Kappa): the rate of change of the derivative price with respect to changes in the volatility of the underlying asset.

Rho: The rate of change of the derivative price with respect to the change in interest rates.
Other Greeks exist but are less universally known. Some traders of more exotic options are interested in the
second order derivatives (i.e. the rate of change of vega with volatility). The most common of these second derivatives
is, of course, Gamma which gets a Greek of its own.
♣ Financial derivatives are instruments that derive their value from other financial insturments, i.e. and option on a share. See mutant's w/u on the derivative node.