Since Japan is such a large country, and many enticing destinations are separated from the main island by water, domestic flights are quite popular in Japan. Due to the recent appearance of a bunch of new Low Cost Carriers on the market, prices have plummeted and you can now fly around for the price of a pair of jeans. In fact, air tickets are often even cheaper than those of the Shinkansen bullet train!
Although flights might be cheaper, somewhat closer destinations may be faster reached by Shinkansen. However, when it comes to distances like Tokyo-Hiroshima (half the country) or farther, you're probably better off by plane. Also, the many islands are easily reached by air plane, such as tropical Okinawa and snowy Hokkaido.
For an up-to-date list of the possible destinations, please check the respective airline websites. I've listed the most common ones below:
And the LCCs operating in Japan: