I have population data matched with polygons and a set of points that I have drawn buffers around. Assuming that the population within each geographical area, i.e. the polygons, is evenly distributed, how do I calculate the population within each buffer in ArcGIS? Some of the buffers are overlapping.
If you need I could send you the data.

Answer
This can be accomplished with an Intersect, followed by a Field Calculate, and then finally a Summary Statistics. Make sure that your buffer feature class has a unique ID field.
Before getting started, you will need to add a field (name Polygon_Areas, type Double) to your population polygon feature class, and then field calculate it, using Shape_Area as the input. This will create a field with your areas that can be carried over in geoprocessing.

Now perform an intersect with your population polygons and your buffers as inputs.

A new feature class is created. Add a new field to it - maybe Area_Pop, type Double.
Field Calculate this new field in your intersect feature class:
Population field * (Shape_Area / Polygon_Areas)

Finally, you can perform a summary statistics on your intersect feature class. Use the Area_Pop as your statistics field and chose SUM as your type. Add the unique ID field for your buffer feature class as your Case field.

Voila. You can perform a Tabular Join from your new table back to your original buffer feature class and add the SUM_Area_Pop field to a new field with a field calculate.

Good luck!
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