Here's the super-short version:
The Invisible Man doesn't get along with anyone in society: he didn't get along with anyone before he was invisible and now things have only gotten worse. He thinks the laws shouldn't apply to him and eventually he gets killed by a mob. Epic fail, Invisible Man.
Here's the slightly longer version:
The Invisible Man starts with a stranger arriving at the town of Iping. He's a private guy, which is a problem when you live in a town where the major export is gossip. The stranger doesn't get along with the villagers, especially the people who own the inn where he's staying. He spends most of his time trying to do something scientific in his room. But eventually – after the villagers (rightfully) accuse him of robbery – the stranger snaps. He takes off all his clothes and reveals that he's – wait for it – invisible! (Yeah, we know, it's in the title.)
The Invisible Man fights the village and flees, leaving his important scientific notes behind. To get them back, he forces a homeless dude named Marvel to help him. They go back to Iping and get the Invisible Man's stuff, but the villagers attack and craziness ensues. The Invisible Man beats them to a pulp and wreaks some major havoc.
At another town (Port Stowe), the Invisible Man steals money and drops it into Marvel's pockets. Like the lousy sidekick he is, Marvel runs away to Burdock, money in hand (or in pocket, we guess). The IM tries to kill Marvel, but a bunch of people at a bar fight him off; one person even shoots him, but it's just a scratch. The Invisible Man takes shelter in a house that happens to be owned by an old college friend named Kemp, and this is where we learn that our not-so-hero's name is Griffin.
While staying in his digs, Griffin tells Kemp his back story story, which is several chapters long (and we mean long). Here's the gist: he was poor and he wanted to study invisibility (as most young people do), so he stole money from his father, who then committed suicide (we're not entirely sure why). Finally Griffin figured out the invisibility thing and proceeded to do a few things: (1) burn down his landlord's building; (2) wander around London; (3) steal from a department store; and (4) put on a ridiculous outfit from a theatrical costume shop and go to Iping to work.
Turns out Kemp had alerted the police to Griffin's whereabouts when he arrived, but when they come to arrest him, he escapes. (Remember, he's invisible, so it's not too tough.) Kemp works with the police to catch Griffin, who in turn, tries to catch Kemp. In the end, a bunch of people in Burdock gang up on the Invisible Man and kill him. As he dies, Griffin loses his invisibility and we get our first glimpse of the Visible Man.
Finally, in the epilogue, we learn that Marvel still has Griffin's scientific notes, which probably have all sorts of cool inventions in them.