First of all:
You will only be able to use ArcGIS with PostgreSQL using OLE DB connections, meaning, you will only be able to read common tables and collumns (you even will be able to read the spatial collumns, but ArcGIS cannot do anything to them,
To use ArcGIS and PostgreSQL + PostGIS (meaning that you need to see spatial data), you will need ArcSDE or ZigGIS.
With both options you can query, edit and analyze data stored in PostGIS, inside ArcMap or other ESRIs tools.
ArcSDE is a middleware provided by ESRI, changing the whole workflow (of installing, configuring a geodatabse, etc) of work and ZigGIS is desktop tool (I mean, only used when ESRIs desktop tools are involved).