This is a hobby shared by people all over the planet. Like other forms of tourism, 'UE' involves finding and learning about sights, going out to actually see (and photograph) them, and talking about it afterward. In this particular instance, the sights are things like abandoned buildings, sewers, tunnels, rooftops, and other areas that are generally thought of as 'off-limits'. It's pretty fun, no matter how you do it or what you call it, especially if you bring a friend or two. As with any other semi-dangerous hobby, being aware of the risks and responsible for you actions are vital.
Much of what I'd say to explain Urban Exploration has already been well said by someone else. If you'd like to learn more, an excellent starting point is Infiltration.org There is a ton of amusing and well-written information on the website.
There are many books published about 'urban exploration', go check out your local library. I would highly recommend "Corporate Wasteland", by Steven High and David Lewis. "Invisible Frontier" by LB Deyo and David Leibowitz is a classic.
I could go on at length about the underground systems of Edmonton or Vancouver, but I'll just sum a few basic facts. Drains (slang for storm sewers) are pipes that convey rainwater runoff from roads and parking lots to the rivers and ocean, often without treatment; generally referring to a trunk pipe big enough to at least crouch through, although many are quite large. There are also sanitary and combined sewers, which I prefer not to enter because they contain raw sewage and dangerous gases. Edmonton has many storm sewer pipes and structures, located all across the city; many of these are quite large, built to drain large amounts of water (Vancouver less so). These sewer pipes are not meant for human passage and there are many dangers present; sewers should not be entered by the unprepared (and never when it rains!).
Over the last decade, I've done a lot of thinking about what 'urban exploration' means to me. My focus continues to be learning about and visiting the more obscure, secret, and historic places in my area. I do this partly for the joy that accompanies discovery, partly to learn more about the city I live in, and also for the mild thrill of adventure. I'm interested in more than just sewers- abandoned buildings, tunnels, bridges, bunkers, hidden spaces- and although not everything makes it onto this website, there are some extra photo galleries. Don't ever stop being curious, don't ever stop looking, and think about where your best memories come from.
This website is a travelogue, where I share accounts of my explorations, express thoughts, and tell everyone what I saw. This website, DrainsofmyCity, has been online since late 2001, steadily growing and changing; this is version 6 or so. I personally design, write, photograph, create, code, and publish everything that goes online (exceptions are noted); please don't use anything without permission. Or maybe this is all a work of fiction.
As mentioned above, Infiltration.org is a comprehensive, excellent resource. The legendary Ninjalicious (aka Jeff Chapman) is credited with coining the term "urban exploration" and many of us followed in his footsteps. The Infiltration zines are, in my opinion, a must-read if you can get ahold of them.
Michael Cook's Vanishing Point contains some very interesting essays about urban exploration and infrastructure, from someone who's seen it firsthand. He looks at this hobby in a much broader societal context, provoking thought and discussions about the hidden areas of cities and how we interact with them. And his photos are stunning.
Although his International Urban Glow website is down, I can still get my fix of amazing drain photographs from Siologen on Flickr. He's the drain-master of five continents, keeper of the list, the Aussie original.