Okay, we all know that steampunk has turned into goths-wearing-brown and terrible things on Etsy but it began life as a pretty fantastic literary genre. Let's talk about it!
For those who don't know, steampunk is "Victorian science fiction", and covers both actual scifi from the Victorian era, and the more modern alternate universe stuff which sprouts from "what if?" questions.
Steampunk-Before-Steampunk-Existed:
Jules Verne and H.G. Wells are the quintessential authors here. For Jules Verne, I'm a big ol' fan of 20,000 Leagues Under the Sea, but your mileage may vary because Verne likes to go on and on for pages about technical details or describing some random shell on the seafloor, so that may or may not be your thing.
Wells is easier to read and is, of course, full of classics: The Time Machine, War of the Worlds, and my personal favorite of his, The Island of Dr. Moreau. His books tend to be short and you can easily read them in one sitting, and they are absolutely worth it.
The Birth of Steampunk:
Steampunk as a modern genre really started with The Difference Engine by William Gibson and Bruce Sterling. It's cyberpunk but set in the 1800s, hence the tongue-in-cheek name steampunk. What if Babbage and Lovelace had actually built that mechanical computer that they came up with back in the 1850s? Victorian alternate history at its finest.
There are other books that started to show up around this time period such as the Narbondo trilogy by James Blaylock, but I haven't read those!
Modern-day Steampunk:
Steampunk has really become ubiquitous as of late, showing up as settings for books, movies, games, music, et al and spawning a DIY movement (think: "steampunking" your computer), as well as the obligatory goth-inspired fashion which involves goggles and brass laser guns and such. There is a tendency for people to throw a bunch of steam-powered machines and cogs and goggles and something and call it steampunk. I prefer to stick with tradition and keep it, well, "Victorian science fiction"!
There are supposedly a couple of gems around such as the Leviathan trilogy; I haven't read them, though! I prefer to stick with the classics, myself.
NOTE: "Steampunk" generally refers to the Victorian/Edwardian era which goes up to about World War I or so. Then you get into dieselpunk. Which is also great.