Europe by train
Travelling through Europe by train is a unique experience. Yes, you can basically go all over Europe, in various countries, only by taking the train. It is also a great way to meet people, either fellow travelers or locals.
Traveling within a country
In many European countries, people mostly travel from one city to another by train. It is often cheaper than flying and the train allows a better access to remote towns.
In France, for example, the SNCF has a monopoly over the rail network. If you do travel often, there are cards available. For a while, I had the 12-25 card, and I sometimes got trains for 25% to 50% off its original price, which is very nice! The card costs 50€ for the year, so you can easily imagine that it becomes profitable after 3-4 travels.
In Germany, I know that the Deutsche Bahn has the monopoly, being the main rail network used to travel around the country. I have never used it, mainly because it can be pretty expensive (bit more than in France). However, in Germany also you have cards such as the BanhCar 25 that grants you 25% discount on routes.