I have often read that the fair value of a stock is the midpoint of the bid and the ask price. That means if bid is 100$ and ask is 101$ then the fair value of the stock is 100.5 $ and if I put in a market order, this is where I can expect the trade to settle.
Is it untrue? I always thought that the trade will take place at either 100 or 101 based on whether I want to sell or buy. So what exactly is the concept of fair value?
None of the above. The fair value is a term used to describe an analytical result of projecting the company's future dividends and profits into a present value. Such estimates are published by the likes of Morningstar, S&P and Value Line. It is quite common for a stock to trade well above or below such estimated fair values.
Fair value can mean many different things depending on the context. And it has nothing to do with the price at which your market order would be executed.
For example if you buy market, you could get executed below 101 if there are hidden orders, at 101 if that sell order is large enough and it is still there when your order reaches the market, or at a higher price otherwise.
If you need to show that the sale/purchase was at FMV, then showing that you made a trade on a public exchange with an unrelated counterpart is enough to establish FMV. However, this is only one of the possible "fair market value" definitions. This is usually used to determine basis or value for tax purposes.
For valuation purposes or general accounting, one specific trade is not enough to establish FMV, and much more research is required.