A map projection is the systematic mathematical transformation used to represent the curved surface of the Earth (an ellipsoid or sphere) on a flat plane. Because a sphere cannot be flattened without tearing or stretching, every projection distorts at least one of four properties: area, shape (angles), distance, or direction. No projection preserves all of them at once.
Why it matters
The projection you choose determines whether your measurements are correct. An equal-area projection (such as an Albers or Lambert Azimuthal Equal-Area) preserves area and is the right choice for computing polygon areas or density. A conformal projection (such as Transverse Mercator or UTM) preserves local angles and shapes, making it suitable for navigation, slope, and large-scale engineering work. A compromise projection balances distortions for general display.
Projections are distinct from, but paired with, a geographic coordinate reference system (CRS). A full projected CRS bundles a datum, an ellipsoid, and a projection, and is identified by an EPSG code.
Concrete example
UTM Zone 31N (EPSG:32631) is a Transverse Mercator projection in metres, accurate for a 6-degree-wide band around 3°E. Web Mercator (EPSG:3857) is a conformal cylindrical projection used by nearly all web tile maps; it badly inflates area toward the poles (Greenland looks larger than Africa) and should never be used to measure area or compute terrain slope.
Common pitfall
Treating Web Mercator (EPSG:3857) or geographic lat/long (EPSG:4326) as an analysis projection. Slope, buffer, and area operations in degrees or in Web Mercator return wrong numbers. Reproject to an appropriate metric, equal-area or conformal CRS before analysis.