A history, background, and rant.
After the 1993 release of id Software's seminal first-person shooter Doom, the first person perspective started to become, shall we say, a bit of a staple.
Let me start over: a first-person perspective is a specific way of presenting a 3D environment on a 2D display, such as a cathode-ray tube. It presents the illusion that what the screen is showing is what someone is seeing; when this sort of display is hooked up to an input device, it presents the most popular method of virtual reality in the world today. First-person shooters, one of the most popular types of games around, rely on this type of display. Compare to a third-person perspective (e.g. Tomb Raider, Grand Theft Auto 3, and so forth).
The aforementioned Doom was not the first first-person shooter, not by a longshot. It did, however, mark the point when games really started using the first-person perspective (and, by extension, 3D graphics in general) as an alternative to the previous staple of the side-scroller. This is simply a result of advancing technology, and really has nothing to do with the relative merits of either 2D or 3D graphics; 2D, sprite-based games are still made today, mostly on handheld systems. Quake, id's later, greater 1996 release, further brought about this change by offering a truly 3D engine which was heavily modifiable, and available to most anyone who wanted to make their own game based on it (that Half-Life, a 1998 release still quite popular to this day, was based on it is quite a testimate to this engine's longevity; of course, Doom was offered to other companies as well). Both Doom and Quake boast a phenominal number of clones (you can tell how long a gamer has been around by whether they use the phrase "Doom clone" or "Quake clone") which were based on their respective engines (remember Blake Stone?). They also offered signifigant multiplayer support, but that lies outside the scope of this writeup.
My point (and you thought I didn't have one) is that, previous to Doom and, to a lesser extent, Quake, the dominant technology for creating a game was through the use of sprites, which really can only be moved laterally (an exception to this is in the SNES, which had a chip allowing the speedy resizing and rotation of bitmaps; see also Mode 7). This was a direct result of the limited hardware available; sprites are very easy to work with. Side-scrolling games (e.g. Super Mario Brothers, Metroid, etc) are one of the simplest ways to use them, and countless games have been made using this format; it's just so simple and elementary. Naturally, Sturgeon's Law (90% of everything is crap) applies: there are a great deal of very crappy side-scrollers for every good one. Technology has now progressed to the point where the first-person perspective is the dominant method of creating a game; the tools and hardware needed are now ubiquitous and cheap. And Sturgeon's Law still applies.
I've spoken with many gamers who both lament the move of games from the 2D to the 3D, and question the usefulness of the 3D format. I say, this move is a great thing, as any new tool is a good thing. The greatest advantage that 3D technology offers over sprites is that one can very easily change contexts; with the usual sprite-based setup, one is always locked in to a specific view of the world. This is not always as much of a limitation as it seems (limitations breed creativity). Just look at one of the greatest side-scrollers: Super Metroid. One is never given the impression that the view is limiting, and yet it is (you could never, say, see something from the top down, or get a nice cinematic sweep of an area). Now look at Metroid Prime, the 3D, first-person conversion of this classic franchise. The gameplay mechanics are converted flawlessly into this perspective, and the ability to present things in a different manner than the first-person view is there: often the view will change (to, say, a sweeping panorama) before a boss fight, to show a sense of drama. Of course, this is merely an option; take Half-Life, for instance, which was highly praised for never removing the player from the first-person perspective, and presented a sense of drama that way. One can also look at the Half-Life mod Natural Selection, which uses the same 3D engine to be both a first-person shooter and a real-time strategy game (and that engine is, at its root, the Quake engine; I told you it was extensible). This is something simply not possible in any other format.
Whether you praise the direction modern gaming is taking or pine for old times, the first-person perspective isn't going away. It has its uses, which are many and varied. The argument is whether, when either option (first-person or something more oldschool) would suffice, is first-person fundamentally better? That's a question for the ages.
Pseudo_Intellectual insists I mention Wolfenstein 3D and Ultima Underworld. Consider it done.