A geographic datum is a specific way of measuring places on earth.
History
Historically, it was very difficult to store and communicate long numbers: to name places in degrees latitude and longitude and have a level of precision suitable for architecture meant writing lots of numbers behind the decimal point.
The idea of a datum helped to solve this problem: instead of having a single reference system for the whole world, you can break the world down into smaller areas with their own local systems. In your town, the local datum might specify that locations are referred to by the number of meters from a certain point.
These local ways of storing places originally had to be manually converted into one another. Luckily, practicing geographers and surveyors can now use software to automatically translate between systems. This software makes it simple to show different datasets in different datums on the same map with a high level of accuracy.
Unfortunately, precise geospatial data isn’t a solved problem. When extremely high precision is required, local datums are still trusted and used more than global systems. Converting between datums often results in loss of accuracy due to the way numbers are represented as floating-point in computer systems.
Technology
Common GIS software like QGIS and ArcGIS contains the code to automatically translate between datums. Under the hood, this is usually provided by the proj4 open-source library.
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