A lot of people, me included, weren't quite sure what was so important about Mass Effect before they played it, so I'm going to talk about why it was (and still is) such a big deal.
I got my (first) 360 not long ago, less than 2 years. I had to decide what games I wanted to get. I researched it a bit, and I decided I had to get Halo 3. (To not do so would be like not getting Twilight Princess for the Wii!) Other than that, I thought I should get one of those action games.. I don't remember what the choices were, but I ended up with Assassin's Creed. Finally, I wanted an RPG. I had the choice between Mass Effect, and Bioshock. (Note: I ended up getting ME on PC, mostly for the benefit of the rest of my family.) At this point, they were a lot alike: incredible, groundbreaking RPG/shooters. I don't quite remember why I chose Mass Effect. It may have been because of the sci-fi setting, maybe because of the developer. (I'd played several Bioware titles, like Knights of the Old Republic and Baldur's Gate.)
I still haven't played Bioshock, and Bioware has come to completely dominate my gaming life. I can't wait for news of the new Dragon Age game, I hang on every word I can find on Star Wars: The Old Republic, and I will die of anticipation for Mass Effect 3.
Why? In short: because the entire concept of Mass Effect is awesome, and Bioware is awesome. For the first, apart from my natural affinity for the setting, it, along with a few other games, essentially spawned a new genre that I have decided to call the Role-Playing Shooter (or RPS, but that doesn't sound right). Bioshock and Fallout 3 have also contributed, but this genre is just as varied as RPGs are. Mass Effect is decidedly different. Story and characters are the dominate aspects of Mass Effect. Bioshock is allegory, a story about a dystopia based on the philosophy of Ayn Rand that happens to be told to a guy with a gun and super powers. (Which look really cool, btw.) Fallout 3 is an open-world RPG (Bethesda FTW), which tend to be light on story. (I think it's because they're too busy building the world) Not that Fallout 3 or Oblivion have bad stories, they're actually both really good, but my point is they aren't what you play the game for. (mostly, anyway)
Mass Effect is a blend of what I would call a true western RPG-- you know, the ones that are essentially based on Dungeons and Dragons, (Speaking of, Bioware made the franchise that is one of the best games actually based on D&D, Neverwinter Nights, and what is IMO the best game that feels like D&D, Dragon Age: Origins) and FPS. RPGs, not necessarily this kind, and FPS are arguably the two most popular genres. I acknowledge that Mass Effect's blend of these is far from perfect, but it was a really good attempt-- and Mass Effect 2 capitalized on that and made it almost seamless.
The first thing that makes Mass Effect different from others like it is the story. It's one of those games that you could reasonably put it on easy difficulty and coast through the combat, just for the main story. You could do the same in an open-world RPG, but that's usually for the little things everywhere, (which are actually one of the weakest points in the original Mass Effect) as opposed to the actual, main story. Mass Effect creates a universe and story that, at least in my opinion, is worthy of the biggest sci-fi franchises out there. The world Bioware has created is, quite possibly, the most in depth universe ever created for a game. They managed to detail the last several centuries or more of the history of the current people, plus the overarching cycle of the entire universe, through dialogue. Then there's the codex- a partly voice-acted repository of almost everything you could imagine in the universe. (Of course, there's dozens of races, thousands of worlds, etc. etc., but it does detail everything you actually see.) There's even several paragraph long descriptions, and, when applicable, histories, of every planet in the entire game. Literally.
Tangent here: I figured out a while ago that I think there are 3 main differences in between Star Trek and Star Wars. 1: The world. Star Trek is the future of this world. Star Wars is in a galaxy far, far away. 2: Humanity's position. In Star Trek, humanity is just another race. In Star Wars they are dominant. 3: The Force. Star Trek doesn't really have any magic, apart from the tech. Star Wars has the Force. Notice that Mass Effect comes out decidedly on the Star Trek side, and the one that is like Star Wars (Biotics = the Force) is far from a central point of the story, like it is in Star Wars.
The second thing is the emphasis on characters. Look at the recent RPGs. (and as far as I can remember, all RPGs) There are western RPGs, and Japanese RPGs (as represented by Final Fantasy). Ignoring the JRPGs, look at the western RPGs. How many of them have companions, ala D&D? All the ones I can come up with from my admittedly biased memory are made by Bioware. Mass Effect takes this to extremes, though. Mass Effect 1 was fairly typical for Bioware, with maybe a few less than normal, with varied classes and races of companions, each with extensive dialogue trees, opinions on your actions, and their own quest.
Mass Effect 2 took this and literally made it the basis of the entire game. There are nearly a dozen characters, most of which have their own recruiting quest and all of which with their Bioware-signature quest-- which are actually an important, necessary part of the game. They are the stars of the game, and Bioware seems to have skimped on the rest of the story in ME2 to focus on them.
In conclusion, Mass Effect is important because it helps to create an entire new genre, but it is so enjoyable because it does it well, and Bioware made it. (Bioware made it = one of the best stories ever. ALWAYS.)