Android Apps, bus time and train time apps, where do they get their information?

Many apps get their static schedule data from GTFS feeds, and a list of these feeds can be found at http://www.gtfs-data-exchange.com/ .

Real-time information systems use a variety of standards. In the US, the market leader is NextBus (Boston, SF and LA among many other small agencies) , which has an open API. Clever Devices also has an API for Chicago and several other cities.

MTA in NYC is using the SIRI open standard, which is gaining traction in Europe as its real-time API implementation (http://bustime.mta.info).

Many transit agencies provide data feeds that include information about schedules, routes, current bus locations, and other information. People write applications that consume this data and present it to the user.

For example, look at the MBTA's developers page, which has information about information available from the MBTA.