A ruler has markings, a straight edge does not. In "straight edge and compass" constructions, you are not allowed to mark the straight edge. Things that are impossible using only a straight edge and compass are possible if markings are allowed (for example, it's possible to trisect an angle using a marked straight edge).
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A ruler is a straight edge
Straight Edge
A straight edge
Generally a ruler is a straight edge, but it can be anything... I generally just use a spare pencil.
A ruler is usually graduated so that it can be used to measure lengths. A straight edge is just that: it has no marking to help you measure lengths.