Even though I'm not Japanese, I ate a lot of Japanese food while growing up. Since Taiwan was under Japanese rule for 50 years, Japanese culture have a lot of influence on Taiwan. Forget what you know about curry, Japanese curry is very different from Indian curry. Most people use Japanese curry blocks to make the curry. You can find these curry blocks at most Asian stores, even in some American supermarkets. The most popular brands are S&B and House brands. My mom uses S&B's Golden Curry. Japanese curry is one of those dishes that you can use whatever ingredients you fancy but carrots, onion and potatoes are a must. It's normally serve with rice but equally delicious with udon.
I'm not posting a full recipe since you can find one in the back of any box of curry blocks. However, I will share my mom's secret for a more flavorful curry. The box instruction will tell you to sauté the veggies with meat until the onion is soften then add the water then curry blocks. My mom's Japanese friend taught her that after sautéing the veggies and meat until the onion is soften, add the curry blocks to the mix then let it cook for about 3-5 minutes (be careful with the heat so you don't burn the curry), finally you can add the water.