You're young and lazy and you've been hired for a summer job at the record store downtown (remember those?) The boss' interest is for her store to make as much profit as possible, long term; this probably routes through selling more records today and keeping customers happy so she can sell more records tomorrow.
Your interest is to collect your $10 an hour, kick back, and do as little work as possible.
The Principal-Agent Problem
There are two problems happening here. First, the interests of your boss (who we call the principal) and yourself (the agent) are not well aligned -- what she "naturally" wants you to do and what you "naturally" want you to do are very different. Second, the boss isn't around all day to check you're doing what she wants you to do (that's the whole point of hiring an employee) -- this leads to assymetric information, where you have information (i.e. how much effort you're putting in to selling records) that the boss would like to have, but doesn't. Together, these two difficulties are the essence of all principal-agent problems.
The boss can try to improve her situation in two ways. First, she could try to tackle the misaligned interests situation. For example, instead of paying you $10 an hour, she could pay you on commission -- you get some percentage of all the record sales you make. Sounds good, right? Now when a customer comes into the store you'll get off your lazy ass and try to encourage him to buy as many records as possible.
The Principal-Agent Solutions
However, it's surprisingly hard to come up with a good incentive mechanism that doesn't create new misaligned-interests problems while trying to fix the old ones. For example, making you work on commission (but with no interest in the long-term viability of the store) might encourage you to do crazy things like pressure customers to buy only on your shift (because you don't get commission if someone you helped buys from the same store on a different day) or to hard-sell to a new customer (succesfully cornering him into buying something from you right then, but losing the store a long-term customer).
And bear in mind, here, that sales is a relatively easy job to find (partial) solutions to principal-agent problems for; there are indeed many salespeople working on commission in the world. But how do you solve incentive problems for a worker with less immediately-measurable outcomes? People who work in back-end roles or on large teams are often doing important work with no direct impact on a single source of revenue, so it's hard to find a way to compensate them that forces their interests to align with the principal's.
The second thing the boss could try to do is to deal with the asymmetric information half of the problem. For example, she could install video cameras in her store ("to improve customer safety") through which she could monitor whether you were actually awake, and on your feet, and not goofing off while being paid. Or perhaps she could do random inspections where she stops by the store unnanounced, or she could survey customers as they leave the store and ask them how the customer service was. However, all these solutions face two big challenges. First, no matter what monitoring scheme she puts in place, you have an incentive to circumvent it and outsmart it. Second, each of these monitoring schemes is costly to the principal, which cuts into the benefit she actually gets from keeping you in check. Ultimately there's no way to completely overcome principal-agent problems.
Uri Bram writes popular non-fiction books with a conceptual approach to mathematical, scientific and analytical thinking. He is the author of Thinking Statistically and Write Harder.