ArcGIS Desktop tool that does for records in a table what "Dissolve" does for features?

You can do this with the Summary Statistics (Analysis) tool. The case field parameter is what you use to "dissolve". From the above link:

case_field [case_field,...] (Optional) The fields in the Input Table used to calculate statistics separately for each unique attribute value (or combination of attribute values when multiple fields are specified).

This sounds like you need to look at the Summary Statistics tool which I find to be a bit of a Swiss Army Knife in that it comes with a Basic (formerly called) ArcView level license and can often be used in place of Frequency (that needs Advanced/ArcInfo) and PULLITEMS (that needs ArcInfo Workstation:-) to reorder fields.