intuition about option values What is the relationship of the option value and the pay off to an option?
What happens if the option value > pay off? or if it is equal to the pay off?

I read that in a put option , the option value is a solution to $$\sup_\tau E[\max\left\{K-S,0\right \}]$$, where $K$ is the exercise price. Does this imply that the possible option values are just 0 or K-S. How important is the option value?
The put option allows you to decide to sell at the exercise price $K$ rather than the market price $S$. 

So if you exercise the put option, in effect it is worth $K-S$ at that point as you could have sold at the market price.  

You will only do that if it is positive. Hence the $\max\left\{K-S,0\right \}$ expression.  

If you could only exercise at a particular future point in time, the option would now be worth the expected future benefit $\mathbb{E}[\max\left\{K-S,0\right \}]$

But with a so-called American put option you can exercise at any time up to a limit, making the value the highest of the future expectations, leading to an option value now of $$\sup_\tau \mathbb{E}[\max\left\{K-S,0\right \}]$$

Typically the option value now is higher than the payoff if exercised now of $K-S$.  This difference is called the time value of the option and reflects the benefit from the future payoff being asymmetrical since it cannot be worse than $0$