At its most basic level, economics is the study of the purchase and sale of goods and services. Consumers and producers buy and sell goods and services at a mutually agreed-upon price determined by the market. But many transactions have side effects that affect third parties beyond the buyer and the seller. These side effects are called externalities.
To understand externalities better, let's imagine a large factory on the banks of a river. This factory does quite well, selling its wares throughout the region. As a part of its production process, this factory produces some small amounts of industrial waste, which it allows to run off into the river. This is the cheapest option for disposing of the waste. (Let's presume, for the purposes of our example, that any environmental laws protecting the river are poorly enforced.)
But imagine now that there exists a sizable community along the banks of the now-polluted river. The products produced in the factory are not relevant to the people in this community. As such, they do not buy anything from the factory, nor do they do any business with the factory whatsoever. Therefore they do not engage in any transactions with the business. Are these people truly unaffected by the factory's business?
Of course not. Even if the neighbors don't participate in any transactions with the factory directly, the transactions have significant impacts on them. Perhaps the children in the community can no longer swim in the river. Maybe the community's drinking water is tainted. If pollution reduces the river's fish population, the livelihoods of local fishermen could be placed in jeopardy. Perhaps the river even starts to smell due to the excess runoff, harming property values in the area.
The economic activity taking place in the factory has created several significant issues for this community. But notice: none of these adverse effects have any impact on the price paid by consumers for the factory's wares. The effects of the polluted water are entirely external to the transaction, hence the term "externality."
Of course, externalities do not have to be negative. In healthcare, for example, the sale of vaccines for contagious diseases provides considerable benefits to the wider community (even to those who do not get the vaccines themselves) in the forms of herd immunity and reduced disease incidence. These positive externalities are sources of incredible value, even though this value is also not accounted for in the sale price of the vaccine.
A PPP designed to address the pollution in the river might try to "internalize" the negative externalities of pollution as well as the positive externalities associated with a clean river. Powerful incentives can be used to ensure that the adverse effects of economic activity are incorporated into the economic equation and that the factory is economically motivated to keep the river clean.