If it adds anything to the following, I'm doing a master's degree in photography and my tutor is the Magnum photographer Mark Power...
In brief, what I look for first in a photograph is content: what is its message, how does it affect me emotionally? Then "structure" - by which I mean how the image has been created and presented, such things as "composition", format, size, and where and how it's displayed. ("Composition" is in quotes because I want to emphasise that it's a more fluid concept than some people think - akin to grammar in language, in that it is in large part an artificial construct, and varies, for example, between societies and over time.)
Something that is easily forgotten but should always be at the fore: the point of a photograph. Photography is simply a means of communication, no different in essence to writing. So, when looking at a photograph, it is essential to explore whether the photographer is communicating their message clearly to their target audience in the way they intend (of course, the audience may be very select, and the message may be deliberately vague). If the intended viewers do not get the message in a photograph, then that photograph is an utter failure. The message does not have to be complex: there's nothing wrong with a photograph that says "Nice sunset". This is something I feel very strongly about.
Like appreciating literature, one can't simply expect to "read" a photograph without some prior knowledge of how images are constructed (analogous to knowledge of spelling, grammar and other conventions in language - you need to be aware of them to appreciate how a writer is deliberately using - or ignoring - them). Some knowledge of how art developed from the start of the 20th century to the present is also needed (how Modernism came about is crucial to an understanding of how photographs are used and understood in today’s society).
How important is structure (e.g. composition) to a photograph? That depends entirely on the aim of the photographer and the purpose of the photograph, and, of course, on the viewer. And you can't divorce the impact of 20th-century art movements on photography - photography and Modernism and Post-modernism (see the footnote at the end of this post for brief definitions without the usual gibberish) developed hand in hand. So, some photographers will take a traditional pictorial approach (taking their cues from European painting before Modernism, typically before 1900), whereas others will be more Conceptual in execution (meaning and intent are all - stuff craft, composition and all that bilge!)
A few questions to think about when you next look at a photograph...
• Why is the truthfulness of a photograph such a big deal? We are properly sceptical of written journalism, but are more inclined to accept that photojournalism depicts reality - despite knowing full well that it has never been any more or less truthful than the written word (and I’m not necessarily implying photo-manipulation - a photographer can lie simply by what they exclude from the viewfinder).
• Why is the debate of photography as craft vs art still ongoing? The National Gallery - the UK’s national art gallery - only recently created a photography department that considers photography as an artistic medium with the same status as any other, such as painting. How does this view of "not art" affect how we perceive a photograph?
• Photographs are ubiquitous in today’s society, increasingly so, and increasingly ephemeral and disposable. Do photographs passively record or catalyse change? And how does this ubiquity and the roles of the photograph in society affect how we perceive and understand them? For example, would Nick Ut’s famous 1972 photograph of the burned girl have the same impact and meaning if taken today in, say, Afghanistan? Would we remember it a half century later?
• We always view photographs bounded within a frame, as if through a window. How does this impact on the way we interpret and see photographs?
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[ Modernism was the attempt by artists to represent the modern, industrial world at the start of the 20th century (e.g. how to represent movement and ideas in art?). Its major characteristic was the rejection of the past and old ways of working and embracing innovation and the new - the "avant-garde". All well and good, and necessary, but it became very cerebral and po faced, and went up its own arse, becoming increasingly out of touch with mainstream society.
Postmodernism evolved midway through the second half of the 20th century. It is a rejection of the Modernistic ideals of the avant-garde and its fetish with the intellectual (especially politics) and "art for art's sake", and embraces the past and popular culture (though you can still be innovative - it's the rejection of boundaries that's the hallmark of Postmodernism). Artists and photographers can actually - shock! - enjoy themselves (e.g. Pop Art) - I'm hard pressed to think of a humorous example of Modernism!
Alexander Rodchenko is a Modernist photographer; Martin Parr is Postmodern. ]