Buying train tickets in Hungary

Do you have to book train tickets in advance in Hungary?

No

Is it possible to buy train tickets on a train

depends

or at Railway stations for immediate travel?

Most of the time yes except some very small stops. And, if you can buy the ticket at the station then you can't on the train without being fined. The fine is 2600 HUF in 2016 and I believe another 500 HUF for missing seat reservations on the trains where that's mandatory. I can't find an English source try Google translating the announcement. The fine is even higher on international trips, 10-30-50 EUR.

Official page:

The ticket sale will start as traffic requires, - but not later than 30 minutes prior to the departure of the train according to time-table. Some ticket sale employees are responsible for other platform duties and so ticket sales may close 5 minutes prior to the departure of the train.

Please, make sure you have enough time to buy your ticket before starting your journey! If you want to be on the safe side, you can avoid crowds and queues by purchasing your ticket in advance.

If you get on the train on such a station or stop where there is no ticket sale, the conductor will sell you the appropriate ticket without any supplementary fee. Make sure you have a valid ticket before you get into a train or train car without conductor (the letters KN indicate it).

Finally, some personal experience: when purchasing a ticket from the machine (there are automated ticket sales now at some places) make sure it's the correct date. The conductor wasn't happy with me last time I've shown a ticket for the next day but accepted it at the end which was nice of him because depending on ticket some are only valid on the day stamped.