Output a table from multiple buffers?

I have a bunch of points in my map. What I would like to do is see how many points are within a certain distance of some specific points. I have about 60 specific points to draw buffers around and that will take a long time. Is there an easier way to get a table of points that intersect the buffers and have a field that tells which buffer it came from?

I feel like this may be a python task but I don't know python :(

Thanks everyone!