Is it possible to buy the ticket on board in German train?

If the train is a Deutsche Bahn service and is operated with a conductor yes, but it costs an additional fee.

But in many local trains of Deutsche Bahn and other services, there is no regular conductor onboard. In these trains riding without a ticket is always considered fare dodging (which costs double the ticket price, or 40 EUR, whichever is greater).

But for Deutsche Bahn you can buy an so-called "Online-Ticket" in advance. You need a credit card to identify yourself in the train, so make sure to bring the credit card with you.

It depends on the type of train, as noted in Aditya Somani's answer. In ICE, IC and EC trains you can buy a ticket in the train without problem (but the ticket will be more expensive than if you bought it online or in the station, see below). In local trains (Interregio-Express, Regional-Express, Regionalbahn, S-Bahn) there's usually a warning printed on the outside of the train: "Einstieg nur mit gültigem Fahrausweis" or something like that (It means "Only get into the train if you have a valid train ticket").

Note that in practice, you often can buy tickets in Regional-Express and Regional-Bahn trains, even if there was a warning to the contrary printed on the outside of the train, so long as you approach the ticket controller before he/she approaches you. I have often done this myself and seen other people do it. However, in theory you cannot, so you may not want to take the risk. If you do decide to do this, make sure to approach the controller rather than waiting for him or her to come to your seat! Note that this does not apply to S-bahn trains, where you will have to pay a fine if you are found without a ticket.

For Deutsche-Bahn trains, the rules are explained in the Conditions of Carriage (in German). In particular, Section 3.9 explains how much you will pay if you buy the ticket in the train:

• In ICE and IC/EC trains you will pay the "Normalpreis" (what you would pay if you bought a ticket at the ticket counter in the station to travel the same day) + a supplement of 7.50 euros.

• In trains of Product Class C (Interregio-Express, Regional-Express, Regionalbahn, S-Bahn) you will pay "Normalpreis" + 10% or at least 2 euros and at most 10 euros.

In practice, in IC/EC trains where you are travelling a short distance and the ticket costs less than about 10 euros, I have found that the 10% supplement is applied instead of the 7.50 euros supplement.

(Of course the fact that the Conditions of Carriage explain how much you should pay if you want to buy a ticket in an S-bahn contradicts the fact that you will certainly pay a fine if you get caught in the S-bahn without a ticket. I suppose only the people who actually made up the rules can explain this...)

That depends on the special train. Here are some questions you need to ask yourself.

Please note that not all payment methods will be accepted on all trains. For example: on DB long-distance services you can only pay by cash or credit card but not with a German bank card (Girocard). On some local trains you can only pay with coins. So check before boarding.

By law you need to be able to get a ticket at the station if you are five minutes at the station before departure or no supplement may be imposed if you approach the conductor directly. In practice this rule may or may not be enforceable depending on your conductor that day.

Is the train named ICE 109x? (where x is any digit)

Sure. You need to pay the price of a regular ticket (Normalpreis), the ICE Sprinter supplement of 11,50€ and the on-board-supplement of 7,50€.

Is the train named ICE, IC, EC, TGV or RJ?

Great. You can buy a ticket on board for the regular price (Normalpreis) of the journey plus a supplement of 7,50€.

Note that some trains for example in Brandenburg or north western Lower Saxony accept regional tickets. You can't buy those on board the train, but you'll still get the pricier long-distance ticket.

Is the train named CNL or EN?

You can get a ticket and night train supplement on board for an additional supplement of 7,50€ if there are still tickets available. Talk to the conductor before boarding.

Is the train named THA?

This train is not operated by Deutsche Bahn. You can get a ticket on board for a 25€ supplement. You need to go to the conductor before or directly after boarding.

You can't get a ticket from a Deutsche Bahn counter or ticket machine.

Is the train named X?

This train is not operated by Deutsche Bahn. You can get a ticket for a special price which is higher than the one in advance.

You can't get a ticket from a Deutsche Bahn counter or ticket machine. There are some other machines selling those tickets.

Is the train named HEX and you are between Genthin and Berlin?

This train is not operated by Deutsche Bahn. You can get a ticket on board for the regular price.

You can't get a ticket from a Deutsche Bahn counter or ticket machine.

Is the train named HKX?

You can get a ticket for a special price which is higher than the one in advance.

You can't get a ticket from a Deutsche Bahn counter or ticket machine. There are some other machines selling those tickets.

Are you traveling by any other train?

This is a local service which may be operated by any company.

The rules are much to complex to list here, would fill books (or rather do fill books), and differ by region, train operator, and even station you boarded. If there is a ticket machine at your station you most likely can't get a ticket on board the train. On some lines you can buy tickets at the conductor or at a machine inside the train but those are rather rare nowadays.

Of course you can board the train and pay the double fare or 40€, whichever is higher. If you do this to dodge the fare, this is an offense under the German criminal code. If you do this in good faith it is not.

There are only few trains with ticket machines and they are not always working.

But you can download the DB-App (DB Navigator) and buy a ticket with your smartphone. The problem is that smartphone tickets cannot be booked for journeys within local transport associations. So if you need a ticket for a short trip, you need to download an app from the local transport association (if available).