The volatility smile or volatility skew is the variation of the implied volatility of an option with its strike price. The term volatility smile comes from the typical shape of the curve when a graph of implied volatility against strike price is constructed.
The variation of implied volatility with strike price is not consistent with Black-Scholes. Variations of Black-Scholes have been developed (using different probability distributions, for example), can account for this.
The inconsistency with Black-Scholes is also a reason not to use implied volatility as an estimate of the actual volatility of the underlying security: it varies with which option you use to calculate it! Although this problem can also be solved by using a different formula, there is no one formula that has been generally accepted as correct (or near enough to be correct).
Volatility also varies with the expiry date of options: volatility has a term structure. The term structure of volatility and the volatility smile can be graphed together as a volatility surface. Analysing the shape of this is of great importance in options trading as it can identify mis-pricings.