Merging Attributes of points into polygon using ArcMap?

I think this classic point-in-polygon overlay operation is best handled using the Intersect_analysis tool wich:

Computes a geometric intersection of the input features.

If you input a feature class or shapefile of points, and another with polygons, then the output is any points falling within a polygon have the ID (and optionally attributes) of that polygon added to then end of its attribute table.