Japanese curry belongs to the group of typically Japanese foods that have origins in European cuisine, called yohshoku. Curry is tremendously popular in Japan - it's on the menu at just about every 'family' restaurant and department store restaurants, and there are curry-only restaurants as well as ones that specialize in high class yohshoku in general.
Japanese curry, called curry rice (or kareh raisu) since it's always served with rice, is not much like the curries from India, Thai or other places with better known curries around the world. The best way to describe it is probably to say it's like a English style stew with curry. (It's not at all like the curries you get in modern Britain, which are firmly in the Indian or Pakistani curry families.)
If you've ever been to a Japanese grocery store, you've probably seen the blocks or bags of curry base taking up an inordinate amount of shelf space. Competition amongst curry base makers in Japan is fierce. The bases are pretty convenient to use, but these days I use them less and less, since I discovered that making curry properly from scratch is not that much more effort than making curry with a readymade curry base. Commercial curry bases contain things like sugar or corn syrup as ingredients, plus some of them use mystery fats (always check the ingredient lists). I add sweetness just via the vegetables, especially a huge mound of slowly sautéed onions.
Either way, to get the most flavorful curry takes a long time. This is definitely a slow-cook meal.
This recipe for beef curry can be adapted to other kinds of meat, or to vegetarian options too. I've included instructions for using a store bought curry base as well as making your own curry roux base.