First one is a neat collection of adventures featuring Batman and Robin hunting down Gotham's villains, each villain in their own unique environment. A real love-letter to Burton's Batman in both music and visuals. [spoilers]and in the second half you get to play as the villains.[/spoilers] My personal favorite of the three.
The second one is probably the weakest of the three, and scraps most of the villains in favor of a Joker/Lex Luthor bromance. Superman plays a big role along with Batman and Robin, and the rest of the League shows up at the very end. The best part of the game is getting to explore the open-world Gotham freely in the Batmobile and other Bat-vehicles, even if it is a little small. Again, I like this Gotham because of the Burton influence (compared to something like the bland generic Gotham in the Arkham games)
The third game in the trilogy is much more of a Justice League game than a Batman game. Brainiac is the villain, and there's a huge focus on the Lanterns as well. It's not really a Batman game at all, considering Batman gets about equal representation as the rest of the League. It's very reminiscient of Lego Marvel (which is also great, despite me being much more of a DCfag) minus the open world. It's a shame Gotham doesn't make an appearance outside of one level, but it's also visit other locations as well, including the Lantern planets.