Simply put, "Resolution" is a term used to describe the potential for detail in an image. The term is used throughout cinema, from lenses, cameras, sensors and images through to projectors and other displays. However, resolution is just one factor contributing to our perception of detail.
Our overall perception of detail depends not just on the finest image features, but also on how the full spectrum of feature sizes is rendered. With any optical system, each of these sizes is interrelated. Larger features, such as the trunk of a tree, retain more of their original contrast. Smaller features, such as the bark texture on a tree trunk, retain progressively less contrast. Resolution just describes the smallest features, such as the wood grains, which still retain discernible detail before all contrast has been lost.
A key reason why higher resolution imagery ends up looking better therefore isn't just because finer detail becomes visible but also because its benefits cascade to features at coarser sizes. For example, having a higher resolution could cause the wood grains to become rendered with as much contrast as the bark was previously. Best of all, these benefits apply even if an image isn't used for display at its full resolution, since higher contrast, large-scale features are retained.
However, one can always have too much of a good thing, and resolution is no different in this regard. If the resolving power of a sensor is pushed too near its pixel resolution, then unsightly digital artifacts will become increasingly apparent. Even worse, these can adversely affect detail at all feature sizes, and drastically reduce video compression efficiency by causing false motion.
Achieving a maximally detailed, natural-looking image therefore requires the right balance of high resolution and low digital artifacts. When done correctly, the benefits translate into applications spanning full resolution theater display all the way down to resized videos on YouTube.
To learn more about about the benefits of resolution, visit the RED 101 Article: https://www.red.com/red-101/understanding-resolution