What is the difference between a single-member district election and an at-large system?
Many political systems, at the local, or city, level feature both single-member district elections and at-large elections. Cities and towns often have city councils, with members elected to represent defined districts within the city or town. To that extent, the system or arrangement is no different form elections at the federal level for the House of Representatives. The House of Representatives, which has its local equivalent in each state, is composed of 435 members of Congress elected by and representative of individual districts. Each congressional district--all 435 of them--has its own elected representatives. Per the terms of Article I of the U.S. Constitution, each state also has two senators, both of whom represent the entirety of their respective states. In other words, U.S. Senators represent all of the districts within their state, and that brings us back to the definition of "at-large" members of city or town councils. While districts, as noted, elect their own representatives, there are also, in many cases, individuals elected with the responsibility of representing the entirety of the city or town in question. These latter individuals run as "at-large" candidates, and are voted upon by the entire polity's electorate. In this respect, at-large members of councils are similar to senators at the federal level.
The difference here is not between single member and at-large. An at-large system can (and often does) have single members. The difference is between a district system and an at-large system. This difference applies mainly to city elections.
In a city with a district system, each area of a city elects a representative to represent it on a city council. In Chicago, for example, there are districts called "wards" that each elect an alderman to represent them. In other cities (like Seattle) there are no districts. Each member of the Seattle city council represents the city as a whole and can be from anywhere in the city.
The at-large system became popular during the Progressive Era because it was seen as a way to break the power of political machines.