In photography the term exposure can be used for many things: it's used to refer to the settings used to create a given image, the act of creating a photo, and the actual photograph itself. These all originate from how the photograph is made, but there is another use for the term: the brightness of the resulting image.
Every choice you make while setting up your photograph impacts the exposure of your photograph. It's a little more complex than simply brightening or darkening a photograph though, because each distinct level of light is affected differently as the exposure is adjusted. If a photograph is overly bright or dark it is considered to be incorrectly exposed, but that doesn't necessarily mean it's a bad photograph. There is much more to the pursuit of a great photograph than making a technically correct exposure.
Throughout the years there have been many methods created to find an "ideal" photograph: from light meters and their exposure charts, to the zone system used by artists like Ansel Adams. What these aim to do is create an image in which the whites are white, blacks are black, and you retain as much detail as possible.
These formulas can be masterfully used if you know what you are doing, but the perfect exposure they create is not always the *best* exposure for your photograph. In reality the best exposure is the one that creates the image you envisioned, whatever that may be.
Exposure and Art
Today, photography is often seen as a very technical and scientific process. Images have to be perfectly sharp, and retain all of the detail in highlights and shadows. While it is good to strive for technically strong photographs, this pursuit can lead to the loss of art in photography.
Deviating from the technically ideal allows you to create unique and interesting photographs that can have a greater impact than a perfectly exposed image.
An underexposed photo can better convey the feeling of an impending storm, even if shadow detail is lost. In the same way, an overexposed image may help you convey the etherial nature of mist in the mountains.
In the end, you need to choose an exposure that best captures your artistic vision, regardless of how "technically correct" it may or may not be.
Choosing Your Exposure
So now you have an idea how exposure can be used artistically, how do you actually decide what type of exposure will be best for your photograph?
The best way to do this, in my opinion, is to listen to your instincts. When you know what the desired mood of your photograph is, whether you are creating one or you are trying to capture the current ambiance, consider how this mood makes you feel. Does it bring to mind images of lightness, liveliness, ephemerality, darkness, etc.
Paying attention to these feelings will give you an idea of the overall atmosphere you want to create. Use this to decide whether your best photograph would use a perfect exposure, or if over or under exposing will create a more impactful image.
Exposure is a key part of photography: not only in how it's made, but also for creating the mood of a photo. The idea of an ideal exposure is flawed because the best exposure is the one that creates the photograph you envision. Striving for technically great images is not wrong, but we must be sure to listen to our instincts while creating. We must never lose sight of photography for what it is: an art.