Pre-rendered background is the process of which the environments you see in the game are not running in real time but instead, were rendered before-hand on different equipment. It usually consists of multiple elements (eg. furniture, trees, vehicles, etc.) rendered into one image and it is then placed in the background of a game level. The developers then add collisions and walk paths to create the illusion that the backgrounds are in 3D space, but in reality they are not.
This technology was created sometime in the early 90's to accommodate the limitation of computer hardware at the time because having to use real-time 3D environments would exceed the graphical memory, causing the game to load longer and sometimes crash.
The Resident Evil and Final Fantasy series (in the original PlayStation era) are known for utilizing this technology, producing memorable set pieces and locations. Rare is also known to introduce pre-rendering in 2D games such as Donkey Kong Country and Killer Instinct.
Pros and Cons
The advantage of using pre-rendered backgrounds is the ability to use graphic models that are more complex and computationally intensive than those that can be rendered in real-time, due to the possibility of using multiple computers over extended periods of time to render the end results. For example, a comparison can be drawn between Still Life (which uses pre-rendered 2D backgrounds and 3D models) and Still Life 2 (3D backgrounds and 3D models); the former's visuals has more fidelity and realism whereas the latter has moving camera angles and "physical" objects.
The disadvantage is generally the low level of interactivity with the environments, fixed camera angles and tank controls, which draws several criticism.
Nowadays, with the latest computer hardware and game engines (Unity 3D and Unreal Engine), both characters and environments can now run in real-time with amazing results, making pre-rendering not as relevant as before. However, some developers still favor pre-rendered backgrounds as paying homage to classic games.