In options pricing models, the most significant (and often only) factor that cannot be fully known is the volatility of the underlying security (how much the price of the instrument you have an option on is likely to move.) By running the models in "reverse" and using the present price of the option as an input parameter, you can calculate the volatility that the underlying should have in order to make the present option price "correct" (ie, to make it fit with the model.) The volatility figure you get is called the implied volatility - the volatility the market implies the underlying should have if the option is priced correctly according to the model. Because the models are imperfect, implied volatility can vary for different options on the same underlying (which it would not do if the models were perfect and the market strongly efficient.)

If not qualified with a model, "implied volatility" will refer to the Black-Scholes model for option pricing.