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I have created a buffer layer and I have a points layer.

How do I count how many points lie in each buffer using ArcMap?

I have been using a spatial join and I know I should probably use the Sum field feature but I don't really understand how that works and if that's the correct solution.

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    Sounds like you want to use 'Count', not 'Sum'. Run statistics using the 'Count' against the Spatial join feature class' attribute that represents the polygon's OID field (or some other field that it unique to each polygon). Commented Oct 9, 2019 at 2:31
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    Perhaps review these previous Qs&As: gis.stackexchange.com/…
    – PolyGeo
    Commented Oct 9, 2019 at 2:59

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You should apply reverse Spatial Join, from point to buffer.

1-Firstly create a field with unique value in buffer polygons (you can use objectID or FID).

2-Than apply spatial join (right click on point layer and choose Join or search it in toolboxes)

3- Summarize the new layer by this field (open attribute table and right click on that field name you will find summarize), result will will have a field *count.

4- You have both layers code by using simple table join you can pass this value to any of layers.

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