It might be worth noting that in the standard Black-Scholes model the implied volatility $\sigma$ is assumed to be a positive non-zero constant. If this were the case then we could simply look at the stock $S$, look at the historical data, and then compute the log-returns on some arbitrary time scale, and then compute the standard deviation, and we would have the volatility.
Unfortunately, the assumption of using a constant volatility is not great, and we find that volatility tends to change over time, (as well as be different on different timescales!). (Furthermore it is not normally distributed, but we can ignore that caveat for now). This leads to stochastic volatility models (e.g. the Heston Model).
Another complication is that the market data is to an extent inconsistent. What I mean by this is that in an ideal world if we took the value of any call option and computed the value of the implied volatility that corresponds to the market price, then we would get the same value regardless of the strike or maturity of the call option. However, in reality we observe that the volatility varies with the strike and maturity, and we instead have a volatility surface. We can then use an extension of the Black-Scholes model and have $\sigma \to \sigma(S,t)$, which is a local volatility model. How we compute volatilities from these surfaces is fairly involved, but a example of how to "calibrate" from market data is the Dupire Equation.
In reality what is normally done is to take some weighted average of a local volatility surface computed from call options (as these are the most liquid). This is then averaged again with put options to improve the consistency of the result. (We could then again take a moving average such as an EWMA over the past and calibrate this further with a stochastic volatility model if we so wished).
Hopefully this demonstrates that in theory there is only one value for the implied volatility, but that in reality this is not the case, and that there are multiple approaches to computing a value for the implied volatility, (each hopefully giving a similar answer which doesn't permit any arbitrage).