There are many variations of oden. Some recipes call for cooking the soup over the course of a few days and other recipes say to do it for a few hours. There are many variations on the ingredients, the most popular ones being:
Hard boiled eggs
Chikuwabu - gluten tubes. Popular in Kantō, virtually unknown elsewhere.
Suji - beef tendons
Kabocha - Japanese squash
Tsukune - fish or meat balls
Tebichi - pig trotters
Ganmodoki - fried balls of tofu mixed with grated vegetables
Atsuage - deep fried tofu
Tofu - mainly in Kansai, usually seared
Bakudan - boiled egg wrapped in surimi
Chikuwa - thick tubes of surimi
Gobomaki - boiled gobo (greater burdock root) wrapped in surimi
Ikamaki - squid wrapped in surimi
Shinjoage - fried seafood paste
You don't necessarily use all of the above ingredients. You can choose the ones that sound good to you, or are easiest to find.
To make oden:
First, make 4 cups of dashi.
1/3 daikon radish
4 boiled eggs
1 blocks of konnyaku
2 blocks of fried tofu (ganmodoki or atsuage)
2-4 fish cakes
4-5 tbsps of soy sauce
1 tsp of sugar
2 tbsps of sake (Japanese rice wine)
Cut daikon into thick rounds. Cut potatoes in half. Peel boiled eggs. Cut other ingredients into large pieces. Add the dashi to a large pot or donabe pot. Add ingredients into the pot. Add sake, soy sauce, and sugar in the pot. Turn down the heat to low and simmer for 40-60 minutes. The longer you cook oden, the better the taste. Add more dashi soup stock and soy sauce as needed.
Oden is usually served in a ceramic lidded pot called donabe. Oden is often served with karashi, which is a Japanese hot mustard. A dab of mustard is added to each bite.