Assassin's Creed and Shadow of Mordor are quite similar in their mechanics. I prefer SoM because it makes me feel much more bad ass to carry out a 100+ hit combo on a group of orcs, hacking and slashing away at their limbs. But the Arkham series is a good deal closer to Shadow of Mordor than it is to Assassin's Creed, especially with it's combat.

In Arkham, the game is third person, like those other games, and it uses a similar combat system as well. You attack by moving (WASD or d-stick) in the direction you want to attack, and you press the attack button. Enemy attacks can be dodged or countered. Combat in the Arkham series also involves a lot of gadgetry, which can be used as parts of combos. Enemies come in the form of different gangs, and each gang has different enemy types, like large slow moving enemies, small fast moving enemies, enemies you can't attack from the front, etc.

A lot of the fun of the Arkham series is discovery, and exploration. The freedom of movement in the games after Arkham Asylum is astounding. Finding riddler trophies, finding locations that are a nod to the Batman series as a whole (the alleyway where Bruce Wayne's parents were shot), solving environmental puzzles using your wits and Batman's detective vision, etc.

The series released in this order:

Batman: Arkham Asylum (2009)

This game is much more "intimate" than the others. The scale is much smaller than the others, as the entirety of the game takes place in Arkham Asylum. The Joker has escaped custody and is wreaking havoc in the facility.

Batman: Arkham City (2011)

In this game, set one year after Arkham Asylum, an entire section of Gotham has been cordoned off and designated "Arkham City", and Dr. Hugo Strange is heading the facility. Batman is incarcerated inside, and must discern the motives of Dr. Strange, and his mysterious "Protocol 10".

Mark Hamill announced that Arkham City would be his final performance as The Joker, however he did reprise the role once more in Arkham Knight.

Batman: Arkham Origins (2013)

A prequel to Arkham Asylum, taking place 5 years prior to the events in that game. This game is a bit of an outlier in the series, as it presents a different Batman than the one we're all accustomed to. The story follows a younger, less refined Batman, who has a bounty placed on his head by Black Mask, drawing eight of the world's deadliest assassins to Gotham City on Christmas Eve while he also encounters Joker for the first time.

This game was also a stand-out from the rest as it didn't feature Kevin Conroy voicing Batman, or Mark Hamill voicing The Joker. Instead, we were given Roger Craig Smith and Troy Baker as Batman and The Joker respectively. This one was probably my least favorite in the series.

Batman: Arkham Knight (2015)

A sequel set one year after the events of Arkham City. The game sees the return of the supervillain Scarecrow, who unites several of Gotham City's greatest villains, including Penguin, Two-Face, and Harley Quinn, to destroy Batman. It also introduces a new villain to the series, the Arkham Knight. It is intended to be the final Arkham game by Rocksteady. Kevin Conroy and Mark Hamill reprise their roles as Batman and The Joker in the game. This is absolutely my favorite game in the series.