PostGIS is an amazing extension to PostgreSQL which makes it possible to manage geospatial data very efficiently. Just in several mouse clicks, after installing the PostgreSQL for your operating system and the PostGIS extension, you will be able to create databases and load geospatial datasets into it for managing later on.
If you are just starting learning about PostGIS, consider going through this excellent tutorial from Boundless (frankly, the best one I’ve seen so far).
You can do pretty amazing things right in your PostGIS database using ST SQL spatial functions. This is handy if your data does reside in the database and you are more comfortable using SQL rather than Python or any other language to access the database and then do the analysis using a geospatial library such as geopandas. Just for your reference, though, there is a Python package GeoAlchemy that can let you work with the PostGIS data and ST functions right from the Python ecosystem. Another Python package, geopandas, can read PostGIS spatial table into a GeoDataFrame so you can work with your data in Python just like you would work with a pandas Data Frame.
For ArcGIS users, it might be also handy to be able to create an enterprise multiuser geodatabase in PostgreSQL database (with PostGIS extension enabled) and then use SQL to do the analysis when you prefer scripting certain workflows without using geoprocessing tools or arcpy scripting. An important note: be sure to create your geodatabase feature classes using a configuration keyword PG_GEOMETRY. Storing your data with ST_GEOMETRY type won’t let you run PostGIS SQL spatial functions on the data.
Please refer to this help page for details on this. You essentially need to create a PostGIS database first (using psql shell or pgAdmin GUI) and then run the Create Enterprise Geodatabase geoprocessing tool. Again, you will be able to load your Esri geodatabase feature class stored in the PostGIS database using the PG_GEOMETRY keyword into a geopandas data frame. Keep in mind though, that to create an ArcGIS enterprise geodatabase in PostGIS, you would need to have an ArcGIS Server license.
I think having an ArcGIS multiuser geodatabase in PostGIS is very appealing because you might have a bunch of other software either serving the spatial datasets (such as GeoServer / ArcGIS Server) or accessing it via SQL in a desktop GIS (such as QGIS / ArcGIS Desktop). Having all the data stored in one place will make it easier to manage and maintain it as you don’t need to propagate the updates among multiple data repositories. Having an Esri complied geodatabase repository with all the powerful features it has to offer (such as geometric networks, linear referencing, geocoding) along with a chance to be able to pull all the data with pure SQL is a very agile concept many would find suitable for their organizations.