bitFlyer Lightning is an exchange (trading platform) where users can trade bitcoins. The exchange rates are determined based on the balance between the supply and demand. Users can sell JPY and obtain bitcoins in exchange, or they can sell bitcoins in exchange for JPY. The practice is similar to retail foreign exchange (FX) trading.
In the bitFlyer Bitcoin Market, your trading partner is bitFlyer. As a customer, you can buy bitcoin from bitFlyer or sell bitcoin to bitFlyer. On the other hand, the bitFlyer Lightning Exchange is a trading platform that bitFlyer provides for you to trade and perform transactions with other users (customers).