You simply take the next train. If you have savings fares which are bound to a specified connection, you need to obtain a written confirmation from the conductor on the delayed train (i.e. BEFORE the change) that the train is delayed, then you can use the next available train without any extra fees.

If you miss a train in Germany, your options depend on the type of ticket you purchased from Deutsche Bahn (DB). If you have a "Flexpreis" (Flexible) ticket, you are not tied to a specific train; you can simply hop on the next available train on the same route on the same day without any penalty. However, if you have a "Sparpreis" or "Super Sparpreis" (Saver) ticket, your ticket is "Zugbindung," meaning it is valid only for the specific train number printed on it. If you miss it due to your own delay (like sleeping in), the ticket is technically void, and you must buy a new one at the current "day-of" price. The big exception is if you miss a connection because of a delay caused by Deutsche Bahn (e.g., your first train arrived late). In this case, your "Zugbindung" is legally lifted; you should go to the "Reisezentrum" (Travel Center) or find a conductor to get your ticket stamped, allowing you to take any subsequent train to your destination for free.