Welcome to the Madhouse.
Story:
Arkham Asylum begins with Joker's takeover of Arkham Island. Batman has to work his way through the island trying to stop Joker, and over the course of the story, you can really see how Batman is pushed to his limits. Joker is a great villain, always taunting Batman and growing noticeably more desperate throughout the story. Perhaps the story's only flaw is that some players may not be familiar with all of the key players. It uses some of the lesser known villains instead of sticking with only the popular ones.
Gameplay:
Exploration is pretty straightforward. You'll head to a location, occasionally taking down goons or using a gadget to bypass obstacles. The combat is fast and fluid. Once you get the hang of it, you'll be taking down goons with ease. The other side of the game is stealth, which is necessary to fight armed guards. You can hide on gargoyles and swoop down on enemies, sneak up on them, or lure them into traps. The stealth really makes you feel like the predator, not the prey. While very easy, the detective segments and minor puzzles are also a nice addition. Overall, the game really makes you feel like Batman.
Graphics:
Arkham Asylum looks great. Its dark, atmospheric visuals really contribute to the experience. Characters look believable and the Asylum looks suitably eerie. Also, Batman's suit takes (scripted) visual damage throughout the game, which is a nice touch.
Replay Factor:
I only rented the game, so I can't say too much about this, but my run (on Normal) lasted around (I'm guessing) 10 hours. Finding all of the hidden riddles, beating the game on hard, and completing all of the challenges could significantly lengthen your play time, but if you only want to play the main story, you could be in for a short experience. The combat and predator challenges basically focus on the combat and stealth sections of the game, however, so if you enjoyed the main game, you'll probably enjoy them. Unfortunately, you unlock more challenges by completing the Riddler challenges, so if exploration isn't your thing, you'll either have to suffer through the riddles or pass up the extra challenges all together.
Overall, the only flaws I found in Arkham Asylum were a couple of crashes and lip-sync glitches, the tendency of detective vision to make things a little too easy at points, villains that may be unfamiliar, and a story that could stand to be a little bit longer. If you're thinking of taking a trip to the Asylum, however, you're in for a Batman experience like no other.