There are numerous countries in Europe from which you can come easily to our capital. Or not so easily.
This is useful map when you plan your trip to Hungary. It shows the European railway connections, or more exactly, the lines that takes you to Budapest directly, without changing.
As you can see there are a great number of cities from which the trains run to Budapest on daily basis, there trains running only on certain days and there are period ones.
You can download the map here.
For the actual international timetable click here.