How does "buy low, sell high" relate to supply and demand? I'm taking introductory microeconomics, and I'm trying to consolidate my understanding by looking at some real world examples.
In a perfectly competitive market, there are many consumers and suppliers who are price takers. They are constrained to buy or sell at the accepted equilibrium price due to many competitors.
Therefore, we should expect $P^* = P_D = P_S$ for a certain good or service in a perfectly competitive market. However, having been to a foreign currency exchange at an airport, this does not seem to be the case! They are willing to buy an item (eg: USD) at a lower rate, but they sell it at a higher rate. So it seems that $P_D &lt; P_S$ for the traders (looks similar to a tax wedge), assuming they are both buyers and sellers of the US dollar in the same market.
This example confuses me, as there seems to be two prices instead of one. In the textbook, if I were to buy $Q^*$ items at $P^*$, I should be able to also sell $Q^* $ items at $P^*$, not $Q^*$ items at $P &lt; P^*$. Other examples of this can be seen whenever sellers and buyers are allowed to post their own listings; there always appears to be a price gap.
How do I explain these real world examples using the concept of supply and demand in a perfectly competitive market (and on the graph)?
In the comments @henry already provided you with the correct conceptual answer, I will try to offer some extra intuition and way how this could be modeled. Currency exchange is a combination of a retail market with service market so here if you would want to visualize it with supply and demand you would have to use two diagrams for both parts of the market.
If that still sounds too exotic just imagine classical retail. A supermarkets will first purchase its produce (let us say carrots) from carrot farmers. This will be separate market where supermarkets are the buyers and where they jointly together form demand and where farmers are suppliers. On this 'market for carrots from farms' there will be supply and demand and price given by the equilibrium.
However, once supermarkets acquire carrots then they will sell them on second retail market to regular consumers. On this second market, let us call it 'market for retail carrots' supermarkets are the suppliers and regular consumers are part of the demand. This market will have its separate price that might be different from the 'market for carrots from farms'. This however does not mean that market is not competitive because supermarket has additional marginal costs for warehousing those carrots, if they sell them just for price that covers these additional cost the market would be perfect (of course, in real life the market might not be perfect, but the point is  you can't just say it from the fact that there is a price differential).
In the same way the exchange window at an airport can be thought of operating at two distinct markets. At a market where it buys currency from individuals where the exchange windows are the customers and part of demand and people are the suppliers, and then second market where the exchange window operates as a retail and act as a supplier and people who buy currency are the buyers.
Again this is all consistent with perfect competition. Consider the following simple model:
Exchange window sells USD at price $P_H$, but buys currency at $P_L$ and exchange window additionally has to pay the person sitting at the window wage $w$ for facilitating intermediation (let us assume that wage is paid per unit of currency exchanged to simplify calculation).  Let the quantity here be the number of USD exchanged ($Q_{USD}$). Now this can be fully consistent with perfect competition. Let us suppose market is perfectly competitive so $P_H$ is treated by firms as being exogenously set (e.g. see any introductory economics text such as Mankiw Principles of Economics). This means that the profit function of this exchange window will be given by:
$$\Pi = P_H Q_{USD} - P_L Q_{USD} - w Q_{USD}$$
Now take the FOCs and we get:
$$ P_H - P_L - w =  0 \implies \underbrace{P_H}_{\text{price}} = \underbrace{P_L + w}_{\text{Marginal Cost}} $$
Hence, here we have literally a textbook example of  perfectly competitive market, where $P_H&gt;P_L$ (e.g. a situation where firm buys low sells high). Yet there is no additional markup as price that the firm provides is exactly equal to marginal cost so the markup $\mu=1$ and we have perfectly competitive outcome.
The intuition here is that for the exchange window buying currency from people is part of their cost, but then they have other costs (e.g. wage in example above and in real life also capital costs etc) since the exchange window is in business of providing intermediation service. In perfectly competitive market price at which firms supplies product will be equal to marginal cost, but here the marginal cost equal not just the cost of accruing the currency from people in the first place but also marginal cost of intermediation.
Lastly note, that this is of course not enough to prove that exchanges are competitive, rather this is proving the opposite. Namely, a difference between price at which firm buys product and price at which it sells product in itself does not tell you anything about whether market is competitive or not. $P_H&gt;P_L$ on both competitive and non-competitive markets. Here you would have to add more reasoning to argue it is competitive, like in a question you mention that when it comes to exchanges there are usually many firms offering the same service and so on. Or you could actually try to empirically estimate marginal costs of the firm and see if indeed firm just sells its product at marginal costs or if the prices are higher indicating both presence of markup and market power.