$match vs $filter, $project vs $map, $group vs $reduce

Four pairs of operators that get confused constantly. In each pair the difference is the same question: does this act on documents, or on the elements inside one document?

$match vs $filter

$match

Filters documents.

$filter

Filters elements inside an array.

$project vs $map

$project

Transforms a document.

$map

Transforms every element inside an array.

$group vs $reduce

$group

Combines multiple pipeline documents.

$reduce

Combines/reduces elements inside one array.

$map vs $unwind

$map

Keep array structure.
Transform elements.

$unwind

Destroy/flatten array structure.
Each element becomes a pipeline document.