Okay, so the Trails series is itself a subseries of a long-running property called The Legend of Heroes. It's a little bit like how the Persona series started life as a Shin Megami Tensei spin-off, but blossomed into its own franchise.
Officially, the Trails games still have The Legend of Heroes in their name. For example, the full title of Trails of Cold Steel is The Legend of Heroes: Trails of Cold Steel — but at least here in the West, fans have taken to shortening the names for the sake of clarity.
Anyway, the Trails titles are role-playing games developed by Japanese company Nihon Falcom (often referred to as simply 'Falcom'). They have a very heavy emphasis on character-driven stories, and feature a turn-based combat system that has steadily evolved over the course of the series.
The biggest strength (and arguably the biggest weakness, from an outside perspective) of the Trails games is that they all tie into each other. All of the Trails games have overarching plot elements, returning characters, and a clear timeline of when events take place in relation to other games in the series.
This massive, interconnected style of storytelling allows Falcom to create a truly dense and engrossing fictional world. It's like an anime that has hundreds of episodes; it can be hard to penetrate such a daunting mass of entertainment, but once you're actually invested, the payoff is almost unmatched.
We would heartily recommend the Trails games to anyone who enjoys Japanese RPGs.