I am looking for a process in ArcGIS that takes a point feature class and groups the points into a user-provided number of clusters.
k-means clustering is exactly what you want. Version 10.1 has k-means clustering... if you are beta program eligible you could try that. You could also try doing your k-means clustering in R and then bringing the results over the arcgis. I think there are also some 3rd party scripts out there for k-means clustering using versions 9.3 and 10.0 too.
If your data is attributed with some sort of "by type" field, you could use the Mean Center geoprocessing tool and set the case field.