The delta, A, of an option or a portfolio of options is the sensitivity of the option or portfolio to the underlying. It is the rate of change of value with respect to the asset:
Speculators take a view on the direction of some quantity such as the asset price and implement a strategy to take advantage of their view. If they own options then their exposure to the underlying is, to a first approximation, the same as if they own delta of the underlying.
Those who are not speculating on direction of the underlying will hedge by buying or selling the underlying, or another option, so that the portfolio delta is zero. By doing this they eliminate market risk.
Typically the delta changes as stock price and time change, so to maintain a delta-neutral position the number of assets held requires continual readjustment by purchase or sale of the stock. This is called rehedging or rebalancing the portfolio, and is an example of dynamic hedging.
Sometimes going short the stock for hedging purposes requires the borrowing of the stock in the first place. (You then sell what you have borrowed, buying it back later.) This can be costly, you may have to pay a repo rate, the equivalent of an interest rate, on the amount borrowed.
The gamma, Y, of an option or a portfolio of options is the second derivative of the position with respect to the underlying:
Since gamma is the sensitivity of the delta to the underlying it is a measure of by how much or how often a position must be rehedged in order to maintain a delta-neutral position. If there are costs associated with buying or selling stock, the bid-offer spread, for example, then the larger the gamma the larger the cost or friction caused by dynamic hedging.
Because costs can be large and because one wants to reduce exposure to model error it is natural to try to minimize the need to rebalance the portfolio too frequently. Since gamma is a measure of sensitivity of the hedge ratio A to the movement in the underlying, the hedging requirement can be decreased by a gamma-neutral strategy. This means buying or selling more options, not just the underlying.
The theta, ©, is the rate of change of the option price with time.
The theta is related to the option value, the delta and the gamma by the Black-Scholes equation.
The speed of an option is the rate of change of the gamma with respect to the stock price.
Traders use the gamma to estimate how much they will have to rehedge by if the stock moves. The stock moves by $1 so the delta changes by whatever the gamma is. But that s only an approximation. The delta may change by more or less than this, especially if the stock moves by a larger amount, or the option is close to the strike and expiration. Hence the use of speed in a higher-order Taylor series expansion.
The vega, sometimes known as zeta or kappa, is a very important but confusing quantity. It is the sensitivity of the option price to volatility.
This is completely different from the other greeks since it is a derivative with respect to a parameter and not a variable. This can be important. It is perfectly acceptable to consider sensitivity to a variable, which does vary, after all. However, it can be dangerous to measure sensitivity to something, such as volatility, which is a parameter and may, for example, have been assumed to be constant. That would be internally inconsistent. (See bastard greeks.)
As with gamma hedging, one can vega hedge to reduce sensitivity to the volatility. This is a major step towards eliminating some model risk, since it reduces dependence on a quantity that is not known very accurately.
There is a downside to the measurement of vega. It is only really meaningful for options having single-signed gamma everywhere. For example, it makes sense to measure vega for calls and puts but not binary calls and binary puts. The reason for this is that call and put values (and options with single-signed gamma) have values that are monotonic in the volatility: increase the volatility in a call and its value increases everywhere. Contracts with a gamma that changes sign may have a vega measured at zero because as we increase the volatility the price may rise somewhere and fall somewhere else. Such a contract is very exposed to volatility risk but that risk is not measured by the vega.
p, is the sensitivity of the option value to the interest rate used in the Black-Scholes formulae:
In practice one often uses a whole term structure of interest rates, meaning a time-dependent rate r(t). Rho would then be the sensitivity to the level of the rates assuming a parallel shift in rates at all times. (But see bastard greeks again.)
Rho can also be sensitivity to dividend yield, or foreign interest rate in a foreign exchange option.
Charm The charm is the sensitivity of delta to time.
This is useful for seeing how your hedge position will change with time, for example, up until the next time you expect to hedge. This can be important near expiration.
The colour is the rate of change of gamma with time.
The Vanna is the sensitivity of delta to volatility.
This is used when testing sensitivity of hedge ratios to volatility. It can be misleading at places where gamma is small.
Vomma or Volga
The Vomma or Volga is the second derivative of the option value with respect to volatility.
Because of Jensen's Inequality, if volatility is stochastic the Vomma/Volga measures convexity due to random volatility and so gives you an idea of how much to add (or subtract) from an option s value.
The above greeks are defined in terms of partial derivatives with respect to underlying, time, volatility, etc. while holding the other variables/parameters fixed. That is the definition of a partial derivative. But, of course, the variables/parameters might, in practice, move together. For example, a fall in the stock price might be accompanied by an increase in volatility. So one can measure sensitivity as both the underlying and volatility move together. This is called a shadow greek and is just like the concept of a total derivative in, for example, fluid mechanics where one might follow the path of a fluid particle.
References and Further Reading
Taleb, NN 1997 Dynamic Hedging. John Wiley & Sons Ltd
Wilmott, P 2007 Paul Wilmott Introduces Quantitative Finance, second edition. John Wiley & Sons Ltd
-  Here derivative has its mathematical meaning of that which is differentiated not its financial meaning as an option.