The amount of light projected at the screen by a projector is measured in ansi-lumen but we’ll simply call it the lumen output. The actual number listed in a projector’s spec sheet is about as meaningless as can be, as all it says is that a 2000-ansi-lumen projector has a higher lumen output than a 1000-ansi-lumen projector. It however doesn’t say anything about the quality of the image or whether a 2000-ansi-lumen projector is better than a 1000-ansi-lumen one. For every single projector the quality and the readability of the projected image depends on the amount of ambient light and the size of the image we’re projecting.
If we're for example projecting in a darkened room, the 2000-ansi-lumen projector won’t have an advantage as the 1000-ansi-lumen projector will have sufficient lumen output under these conditions to create a crisp and clear image. The reason why is simple, because there’s no, or not enough, ambient light to affect the projected image. The other way around the lumen output does matter however, if we’re for example projecting in a well lit room, for example a company board room, then more lumens will mean a better defined image, so the 1000-ansi-lumen projector is at a disadvantage here.
Fig 2. The left projector clearly has too much lumen output, saturating the image, the right one is alright.
However we must not forget that in order to be able to make any comments about the readability and quality of the projected image we need to factor in the size of the projected image. If we, for example, take another look at the 2000-ansi-lumen projector and have it project an image of 2 x 1.5 meters and we enlarge the image size to 4 x 3 meters, then the surface area of the projected image has been multiplied by four and thus the brightness of the projected image is just a quarter of the original 2 x 1.5 meters image.
So, in a nutshell, the lumen output of a projector is important, but only if you factor in the lighting conditions of the room it’ll be used in and the size of the image that you’ll be projecting. In the case of a home theatre, lumen output isn’t really an important factor, as we’re always projecting in a darkened room and at a known image size. Most home theatre setups use a projector with a lumen output around 500 ansi-lumen to not saturate and reduce the contrast of the projected image.