What makes Pareto efficiency important is that almost everyone would agree that society should avoid situations that are not Pareto efficient. That is, when something could be done to make at least one person better off without hurting anyone, most people would agree we should do it.
Economists often describe inefficient situations (those that fail the Pareto test) as "leaving money on the ground." To see why, imagine that someone walking along a deserted beach finds a $20 bill on the sand. If she picks it up, she's better off and no one else is worse off. Leaving the bill on the sand to wash out to sea would be silly. You might be tempted to reply "Wait, isn't the original owner of the bill worse off?" The answer is "No" because it's important to distinguish between two different events: (1) the original owner losing the bill, and (2) the other person finding it. The original owner definitely became worse off when he lost the bill. However, once the bill is gone, he'll be the same whether someone picks it up or it washes out to sea.
Pareto efficiency is important because it provides a weak but widely accepted standard for comparing economic outcomes. It's a weak standard because there may be many efficient situations and the Pareto test doesn't tell us how to choose between them. Imagine that two people are walking together along the beach when they find the bill. All of the following are efficient: one person picks it up and keeps it; the other person picks it up and keeps it; one person picks it up and gives $10 to the other; and so on. Who gains from finding the bill is quite different in those scenarios but they all avoid the inefficiency of leaving it sitting on the beach.
Much of economics is concerned with identifying inefficient situations and designing policies and institutions that will promote efficiency and reduce inefficiency. A policy or action that makes at least one person better off without hurting anyone is called a Pareto improvement.
The term is named for an Italian economist, Vilfreo Pareto.