How to perform asynchronous cascade slave replication of PostgreSQL database? This is a feasibility/architectural question (I am no database administrator). Let's imagine a PostgreSQL database running online (like those from Heroku for example).

One would like to maintain a slave copy of this database on a remote PC by connecting to this online database (24/48 hour intervals). Is this possible from a PostgreSQL perspective? If yes, what would be the best Postgres implementation/configuration strategy? What do I need to learn?

The slave copy should catch up on all modifications made to the online database (eventually by specifying some kind of 'until' date if necessary). This slave would be used in read mode only. Ideally, slaves to the slave must be implemented in this asynchronous way too (i.e., in a cascading fashion).
It's Postgres or PostgreSQL, not PostGre.  The method you want is streaming replication or Point In Time Recovery (PITR) which is supported in 9.0 and above.  It's covered in the docs.  Read up on it, and then if you have any questions, come on back with them