Sometimes, those calculations are very clearcut, in the case of valuing an option at expiration, and other times not so much, as in the case of valuing an outofthemoney option on a highly volatile stock that expires in two years.
What is Intrinsic Value?
The intrinsic value of an option contract is its value if exercised today. You essentially subtract the strike price from the underlying asset's current price, and you get your intrinsic value.
For example, stock XYZ is currently trading at $100, and you own the $95 call expiring in 21 days. The intrinsic value would be:
100  95 = 5
What is Extrinsic Value?
The extrinsic value of an option is anything in excess of the intrinsic value. Using the same example as before, XYZ is trading at $100, and you own the $95 call, which is currently trading at $8.00.
We can calculate the intrinsic value by subtracting the strike price from the underlying price, and we get our intrinsic value of $5.00. Now we subtract the intrinsic value ($5) from the current option price ($8) and get our extrinsic value: $3.00.
You might ask why extrinsic value is even a thing. After all, why would you pay more than the intrinsic value? Wouldn't you buy the stock outright instead?
Realize that professional options traders are brilliant and quantitative in everything they can do. The moment someone thinks that an option is selling too rich, there are intelligent and wellcapitalized options traders standing in line to sell it for a cheaper price until the market reaches an equilibrium.
Before we get into some conceptual reasons why extrinsic value should exist, let's break down the fundamental factors of valuing an option contract.
The BlackScholes model takes the following inputs to price an option:
 Price of the underlying stock/security
 The strike price of the option
 Time until the option expires
 The riskfree interest rate you can get from investing in shortterm government bills

The volatility of the stock
Underlying Price and Strike Price
As you can imagine, the current price of the underlying plays a significant factor in the price of an option. You'll pay far more for an option struck at $500 when the underlying is trading at $550 than if it's trading at $200.
Ultimately, the relationship between the strike price and the underlying price matters most. Generally, options traders refer to the "moneyness" of options in three ways:

Outofthemoney: the call strike price is above the underlying price, or the put strike price is below the underlying price. An option is outofthemoney if it would be worthless if exercised today (no intrinsic value)

Atthemoney: This is when the strike price of an option is identical to the current underlying price. For example, if XYZ is trading at $100, the ATM strike would be the $100 put or call.

Inthemoney: Calls with strikes below and puts with strikes above the underlying price are referred to as inthemoney. For example, a $95 call with a $100 underlying and a $105 put with a $100 underlying.
Time To Expiration
Options are finite securitiesthey have a definitive expiration date, after which they are no longer exercisable. For this reason, much of the thought among options traders and academics has been put into valuing the price of time over the years.
Common sense tells us that an option expiring 200 days from now should be worth far more than one expiring tomorrow. The more time you have until expiration, the more time the stock has to move in your expected direction and get your option into the money.
Suppose all options were priced based on intrinsic value. In that case, you'd essentially be able to "freeroll" by buying options with super long expiration dates (200+ days) and simply wait for the stock to experience some upwards volatility to sell them. It'd be free money, which, as you should definitely know by now, the market rarely gives.
Volatility of the Underlying
The volatility, or how much the underlying moves daily, dramatically affects an option's price. An intuitive way to understand why this is the case is to think about two different call options:

An outofthemoney call option expiring one year from today on a mature company in a lowgrowth industry like utilities or tobacco
 An outofthemoney call option expiring one year from today on a growth stock like Tesla in a new industry
To keep things simple, imagine both underlyings are trading at $100, and each call is about 30% outofthemoney.
Which would you rather own? Most would respond with the growth stock, regardless of whether they would want to invest in the company. The simple reason is that a stock like Tesla has massive swings, and the chances of the stock being up 30% or more within the next year are much higher than that of a boring tobacco or utilities.
As a result of this, volatility has a price.
Bottom Line
There are very good reasons for extrinsic value to exist. Markets are pretty efficient at pricing options and won't give you a freeroll in the form of free optionality by letting you buy a Tesla call for the same value as the equivalent call in a utility company.
To summarize:

The intrinsic value would be the value of an option based on its "moneyness" if it were to be exercised today.

Extrinsic value is the value of an option based on all other extraneous factors unrelated to its intrinsic value, like the underlying's volatility and time to expiration.
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