TITLE: Can someone explain curvature in simple terms
QUESTION [5 upvotes]: I am studying differential geometry but am having a hard time picturing curvature. Can anyone explain it to me in simple terms, perhaps with any diagrams. As simple as possible!

REPLY [4 votes]: Curvature is how fast the direction is changing as a point moves along a curve.
In physical space, curvature is measured in radians per meter or radians per mile or degrees per mile, or the like.  If you move one meter along a path, by how many degrees does your direction change?  Divide the change in direction, in degrees, by the distance, in meters, and you've got the average curvature, in degrees per meter.  Then take the limit as that one-meter-long curve shrinks to a point, and you've got the curvature in degrees per meter at that point.
In pure mathematics, one often regards distances as mere numbers instead of multiplying a number by a unit of measurement.  Then the curvature becomes a dimensionless number.