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Arcmap 10.1: I edited the coordinates within a shapefile. They are long/lat but the point will not move to that new coordinate on the map. I know this is probably a simple fix but for the life of me I am drawing a blank. I've tried clicking on the Longitude column in the attribute table and 'calculating geometry' but still nothing occurs.

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Use field calculator expression arcpy.Point(!PointX!,!PointY!) on field Shape. – FelixIP Jan 6 at 21:32

Start editing that shapefile, double click on said point, right click on point and "move to" and it will give you an option to type in the x and y coordinates for where you want to move the point to. the drop down also gives options to change to DD or meters etc.

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You are only changing the attributes and not the actual location of the point. The X and Y columns contain only attribute data and not the actual shape/spatial data of the point. If you want to move the point, start editing, click on the point, and drag it where you want it to go.

Alternatively you can add new points at a specific X,Y to the shapefile like this :

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