I start with fish, since most (American) tourists eat German meat dishes.
Forelle Müllerin - trout with almond butter.
Scholle Finkenwerder Art - plaice Finkenwerder style - with bacon bits
Büsumer Krabbenbrot - Büsum-style shrimp sandwich: German black bread (Pumpernickel) with shrimps and fried or scrambled egg
Heringsmarinaden - marinated herring. Herring is abundant in the North Sea and the Baltic Sea. We eat salted herring (Matjes), raw herring and fried herring, plain, with rolls or as salads. Just a few examples:
Matjesbrötchen - salted herring roll
Brathering - fried herring, marinated
Rollmops - marinated raw herring, rolled up
Heringssalat - herring salad (often with red beet)
Seezungenröllchen - Dover sole rolls
I stop here. Now, vegetable dishes (or dishes where vegetables play the dominant role).
Semmelknödel mit Pilzen - bread dumplings with mushrooms
Leipziger Allerlei - peas, carrots, asparagus, morel, cauliflower and crayfish
Grünkohl - cale, extremely popular when in season (in wintertime), usually served with bacon, smoked sausage and cured pork
Kohlrouladen - cabbage rolls, stuffed with minced meat
Now, the meat dishes.
Königsberger Klopse - meatballs, Königsberg style with anchovys and capers, in a creamy sauce. One of the most popular German dishes.
Rouladen - beef roulade
Hühnerfrikasse - chicken stew
Kaninchen - rabbit
Rehrücken - venison (rack of roe deer)
Himmel und Erde - Sky and Earth, blood pudding with mashed potatoes and caramelized apple
Mett - raw pork
Now, the one-pot-dishes, stews or rather: thick soups. Made from potatoes, peas, lentils, barley or just vegetables.
Graupensuppe - barley soup
Linsensuppe - lentil soup
Finally, a few desserts.
Kalte Hundeschnauze - cold dog’s snout. A classic for children birthday parties.
Bienenstich - bee’s sting
Frankfurter Kranz - Frankfurt wreath (?)
Maulwurfkuchen - mole pie with banana
Flockensahnetorte - whipped cream pie with lingonberries
I could continue forever…
EDIT:
Thank you all for your comments. Some mentioned that similar dishes occur in cuisines of other countries, like Scandinavia, Switzerland, Austria, Netherlands, Poland, even Ukraine, or France. This is right.
Many recipes are several hundreds of years old and were not invented by a certain chef, but gradually evolved from the products and ingredients which were available in a certain region. Some of these dishes developed before nation states came into existence in the middle of the 19th century. Also, people used to travel, especially between neighbouring countries. So, it is natural that neighbouring countries with a similar climate have the same dishes. Please refrain from claiming that a certain dish is the “national” dish of a certain nation. This would be an ahistoric, nationalist view.
I just listed dishes which are popular in Germany, being aware that they are popular in other countries, too.