This looks like something produced from Density tools in the Spatial Analyst extension.
As that's a web map, it could be as simple as publishing a feature layer using ArcGIS Server or ArcGIS Online, creating a web map and applying a heat map renderer to your layer. How it turns out depends on what your point data look like though.
You will have to take a snapshot of the map you are trying to creat, and give a little more information about how you data points are stored. Are they in an excel spreadsheet? A shapefile? A feature class?
lots of questions, I know. And a picture we can see would help answer some of this maybe. Many here willing to help, but not enough info yet for us to answer.