Doom to Dunia: A Visual History of 3D Game Engines
Quake
Id Software's first truly 3D game engine, the development team went to great pains to ensure the Quake engine ran smoothly without a ton of processing power. One way they did this was by introducing a new way to render maps that purged certain areas from processing that the player wouldn't be able to see. Objects, or brushes as they were called, made up the border of the map and created an enclosed space. The map would then be run through a rendering preprocessor, which would identify empty space inside and outside of the border. It would then discard the back portions of the border. This highly effective technique reduced the amount of polygons usually by half and sometimes by much more.
To further reduce demand put on the CPU, the Quake engine also took advantage of Z-buffering, which put simply is a method for determining which parts of the map are visible and only rendering those sections.
Quake also included 3D light sources, which were added with a second pass of the preprocessor rather than on-the-fly by the CPU, and it also supported 3D hardware acceleration. John Carmack would later release a native port of Quake called VQuake, or Vérité-accelerated Quake, designed to take advantage of the Vérité 1000 graphics chip's hardware features, including anti-aliasing. OpenGL support would also be added, giving Voodoo and PowerVR owners justification for their discrete graphics purchase.
Date Released: 1996
Notable Games: CIA Operative: Solo Missions, HeXen II, Laser Arena, Quake, Silver Wings, Urban Mercenary
Renderware
The Renderware game engine claims a ton of titles under its belt -- over 200 in all -- most of which are for the PlayStation 2 console, but still over two dozen on the PC platform. It's also been used on the GameCube, Wii, Xbox and Xbox 360, PlayStation 3, and PSP, making it very much a multiplatform game engine.

Originally developed in 1998 as primarily a PC-based middleware solution, Renderware, like most of the engines up to this point, largely preceded the GPU, and this may have led to the engine's eventual downfall as a leading API. In an interview with Gamasutra, William "Bing" Gordon, then CCO for Electronic Arts, the publisher who purchased Renderware producer Criterion in 2004, lamented that "Renderware didn't get the next-gen parts that we needed. We actually underestimated Epic early on. They told us, 'We're going to do this, this, and this,' and we thought, 'Eh, it's going to be kind of hard.' We also underestimated our team, then we looked up three months, six months, and nine months later and said, 'Whoops, we underestimated Epic.'"

Before Epic's Unreal engine began to overshadow Renderware, Renderware proved popular for its ability to allow developers to manipulate art and game processes in real time. A developer could, for example, change the color of a character's clothing without altering the underlying code and rendering the scene all over again. This also worked for rudimentary physics, like jumping and moving. If movement looked 'off,' a developer could go in and alter the physics and see the changes in real-time.
Date Released: 1996
Notable Games: Airport Tycoon, Apache Air Assault, Bratz: Rock Angelz, Broken Sword: The Sleeping Dragon, Burnout Paradise, Chinese Paladin 4, Cold Fear, Commandos: Strike Force, Dream of Mirror Online, Evolution GT, Frank Herbert's Dune, G-Nome, kill.switch, Madagascar, Manhunt, Red Jets, Startdon 3, Super-Bikes Riding Challenge, Tony Hawk's Pro Skater 3, 4, and Underground, Trickstyle
Quake II / id Tech 2
Whereas the original Quake game offered hardware graphics acceleration, Quake II offered native OpenGL support right from the get-go. Other features of the Quake II engine, now known as id Tech 2, included colored lighting effects, and a new game model whereby game code was written in C and loaded from a DLL (Dynamic Link Library) rather than the original QuakeC scripting language. In layman's terms, this allowed for both software and OpenGL renders rather than one or the other, so if you didn't own a Voodoo videocard, you weren't necessarily out of luck.

Quake II also became known for its moddability. Because DLLs were also used for the game logic, id could release the source code into the modding community (and it did just that in 2001) while still keeping the rest of the engine proprietary. The engine was also incredibly robust, and savvy developers were able to use it to power full-fledged role-playing games (like Anachronox) and add features like dismemberment (a la Soldier of Fortune).
Ready for a fun fact? The infamous abomination known as Daikatana employed the Quake II engine.
Date Released: 1997
Notable Games: Anachronox, CodeRED: Alien Arena, Daikatana, Heretic II, Kingpin: Life of Crime, SiN, Soldier of Fortune, UFO: Alien Invasion, Warsow