Vega measures the change in an option price due to a 1% change in implied volatility, given that all the other factors which affect the option price remain constant. It is one of the most important Greeks because changes in implied volatility affect greatly the price of an option.
Assume that we have a call (the same is applied to puts) with premium $2.50 and a Vega equals to 0.10. This means that if implied volatility rises 1% the premium will rise $0.10, from $2.50 to $2.60. If implied volatility falls 1% then the premium will decline $0.10, from $2.50 to $2.40.
Notice that a rise in implied volatility augments the prices of calls and puts, no matter which is the directional bias of the market, because high implied volatility translates into high demand for all options, calls and puts and this rises their prices.
Mathematically, Vega is expressed by the first derivative of the option price relatively to a 1% rise in implied volatility and its mathematical formula is:
The same rationale can be applied to Vega relatively to option expiration. The more the days until expiration the higher the time value and the higher the time value the higher is the Vega (obviously all the other characteristics of the comparable options, excluded expiration, are the same).
In a strategy that involves writing of options in order to profit from the deflation of implied volatility, the higher the Vega the steeper will be the decline of the premium value and consequently the profits of the writer.