Students ask me this all the time, so here’s my explanation of the three types of inefficiency.
1. Productive inefficiency. This is a supply-side idea. Mattie and Joe both produce bananas. The bananas are identical. It cost Mattie $12 to produce his last kilogram of bananas, and it cost Joe $10 to produce his last kilogram. They would both better off if, instead of producing that last unit for $12, Mattie paid Joe e.g. $11.50 to produce it. That way Mattie saves 50c and Joe makes a profit of $1.50.
The condition needed to satisfy productive efficiency is that the marginal cost for Company A equals the marginal cost for Company B. If the marginal costs differ, then there are gains from trade from the low marginal cost company (e.g. Joe) selling the item to the company with the high marginal cost (e.g. Mattie).
2. Distributive inefficiency. This is a consumer-side idea. Let’s say that for some reason bananas sell for $1 in the US and $3 in Canada. Then there are people in Canada who are willing to pay, say, $2.75 for bananas but do not receive them. Suppose some American is willing to pay up to (but no more than) $1.20 for bananas. If he sells the bananas to the Canadian for e.g. $2.50, then everyone is better off. Although the American was receiving 20c in consumer surplus by purchasing the bananas for $1, he is clearly better off by selling them for $2.50. (Obviously transport costs have to be small for this to hold.) The Canadian gets the bananas and 25c consumber surplus, so he is also better off. Therefore the $1 and $3 prices generate inefficiencies.
The condition needed to satisfy distributive efficiency is that the marginal benefit to Mr 1 equals the marginal benefit to Mr 2. If they differ, then there are gains from trade from the person with the low MB (e.g. $1) selling the item to the person with the high MB (e.g. $3).
3. Allocative inefficiency. This puts the consumer-side and the producer-side together. Suppose everything regarding productive efficiency (marginal costs for Company A and Company B are the same) and distributive efficiency (marginal benefits for Mr 1 and Mr 2 are the same) is just fine. You can still have inefficiency if the marginal benefits don’t equal the marginal costs. For example, if the costs of the last kg of bananas is $10 but consumers are only willing to pay $3 for it, then producing that last unit is inefficiently costly.
Thus the condition needed to satisfy allocative efficiency is that the marginal benefits (consumer side) equal the marginal costs (producer side).