The Doom engine renders walls by utilizing its Binary Space Partitioning (BSP). The engine renders the walls by traversing the nodes (which represent sectors/subsectors) in the Binary Tree, which draws these subsectors by order of distance from the camera; this means the closest segs (line segments of a wall) are drawn first. As segs are drawn, they are stored in a Linked List data structure. The Linked List is used to clip other segs AND the edges of sprites rendered later on. To reduce the burden on the engine's performance, there is a static limit of 256 segs that may be rendered at once. The excess segs are NOT drawn, leaving visible gaps where walls should be. However, because of the order in which segs are drawn, the gaps are out in the distance where they are less noticeable.
When the engine reaches a solid (one-sided) wall at a particular X coordinate, no more lines are drawn in that area. As this occurs, the engine creates a blueprint (or map) of the areas of the screen where solid walls have been reached. This implies that areas which are currently invisible to the player are clipped completely. Essentially this means that whatever the player is not looking at in the engine is NOT being rendered!
Wall textures are stored as a bunch of vertical columns, which is useful to the renderer since it can render walls by drawing many vertical columns of textures. Since all the walls in the doom engine are drawn vertically, the player cannot look up or down realistically. It is possible to approximate looking up and down via "Y-shearing." Y-Shearing works by increasing the vertical resolution and then imagining a "viewing window" on that space. One can imagine the scene being rendered as a very tall image, in which a sliding window represents the player's view. Sliding the window upwards corresponds to looking upwards, and sliding the window downwards correspond to looking downwards. As the window moves up or down, it creates the illusion that the player is looking up or down. The view tends to become more distorted as the player looks further away from the horizon line.
Y-Shearing image representation