The Legend of Heroes is old-school JRPG with new-school sensibilities and graphics. The old-school elements of the game are its turn-based combat, linearity, long dialogue scenes, a digital pet system and lots of similar things that seem to date from the 1995-ish RPG era. There are no random fighting encounters however. In this game you control Avin in his search for his younger sister Eimelle. They were separated when their cathedral was raided by religious zealots.
A Tear of Vermillion is a remake of the Japanese PC game Legend of heroes IV: Akai Shizuku. The word Gagharv (in the Japanese title of the remake) refers to the chasm that separates the continent Tirasweel, where the games takes place, from the rest of the world. Improvements Bandai made for this remake are completely re-done 3D graphics, new battle interface, and quicker load times.
According to a press release, the translation of the Japanese title in English is Legend of Heroes: Gagharv Trilogy IV - Crimson Tears. A more literal translation would be Legend of Heroes: Gagharv Trilogy - A Drop of Red, though.