Implied volatility, as its name suggests, is the volatility implied (contained or priced in) the price of an option. According to commonly used option pricing models (such as the Black-Scholes model), option prices depend on a number of factors, including among other things underlying price, time to expiration, and volatility. Other factors being equal, the higher implied volatility, the higher the price of the option.
Because the option pricing models are usually mathematically quite complicated (they reflect the fact that the exact relationship between volatility and option prices is also quite complicated), it is not possible to derive a direct formula for implied volatility from common option pricing models. However, you can get to implied volatility by trial and error, which is easy to do in Excel using the Goal Seek feature. The process is explained here: Calculating Implied Volatility in Excel
You can also calculate implied volatility (using Excel Goal Seek) comfortably in the Black-Scholes Calculator. Its PDF guide includes more detailed discussion about it, as well as about vega and the relationship between volatility and option prices. You can also use it to simulate how implied volatility, vega, or option price will behave under different scenarios (e.g. change with passing time or changing underlying price).