Typically, you have a server(s) setup that can render map tiles and maybe cache them locally and you Android device would request tiles from that. There are lots of tools for this. I use mapserver.org for map rendering and mapcache for tile generations and management.
For the routing piece, you would probably want to generate an Ajax request or the equivalent for the Android application back to your server passing the start and end point of the route, then have the ajax handler on the server open a connection to the database and issue an appropriate pgrouting SQL query based on the user request. Then it would format the results as appropriate and return them to the Adroid device where you would then display the response.