That depends on the state and what kind of railroad police you're talking about. In California they are full sworn officers who have general police powers (i.e., they're not limited to operating on railroad property). In other states it's different.
Amtrak police are federal agents with nationwide jurisdiction. That makes some degree of sense when you realize that pursuit might easily cover several states. The crime itself might even cover several states -- imagine someone burglarizing sleeping compartments on the train. When he hits the first one the train is in Minnesota, but by the time he finishes, the train is in Wisconsin. Having federal jurisdiction avoids pissing contests about which state has authority.
The situation is similar for the sky marshals on planes.
I believe they traditionally have dealt with a lot of mail fraud and child pornography investigations.