Definition
Take the Black-Scholes formula, plug in the observed market price, and solve backwards for the volatility input. That number is the implied volatility.
It is the volatility that would justify the price you can actually trade at.
Why traders quote in vol
Option prices move constantly as the underlying moves, even when nothing about the option's value proposition has changed. Volatility is far more stable, so quoting in vol terms strips out the mechanical part.
Saying "I'm bidding 18 vol" is a statement about the thing actually being traded. Saying "I'm bidding 2.40" becomes stale the moment the stock ticks.
It also makes options comparable - a 3-month and a 12-month option at different strikes cannot be compared by price, but can be by implied vol.
Is it a forecast?
Partly. It is the market's risk-neutral expectation of future volatility, which is not the same as its real-world expectation.
Empirically, implied volatility tends to exceed subsequently realised volatility. That gap is the variance risk premium - compensation for bearing the risk of a volatility spike, and it is why systematically selling options has historically made money right up until it catastrophically has not.
Being able to name both halves of that sentence is the mark of a good answer.
The smile
Black-Scholes assumes one volatility for all strikes. Reality disagrees: plotting implied vol against strike gives a smile or skew, typically with out-of-the-money puts priced at higher vol than calls in equity markets.
The interpretation: the market prices in fatter tails and crash risk than a lognormal distribution allows. See the volatility smile.
In the interview
Common question: "An option's price went up but implied vol went down. How?"
The underlying moved in a direction that made the option more valuable - for example a call went into the money - by more than the vol decline reduced it. Price and vol are related but not equivalent, and separating them is the point of the question.
Practise in finance and derivatives.