I am a male in my late 20s. I've never heard of the book One Day is based on, and just went along with my girlfriend to see it (based on seeing the trailer it seemed light and 'inoffensive' at best). I was surprised at how much of an effect it had on me. It is a bitter-sweet story of two people who strike up a relationship while at Uni, but never quite get together. The film then follows them through the next 20 or so years, dropping into their lives on the same day each year.
Despite the first reviewer's opinion that he found the central relationship hard to buy, I found it very believable - it is easily possible for two people to be great friends, find support in each other, and 'light up' in each other's company, but if they were to have a relationship, it just wouldn't work. So, for me, it was completely realistic, I never thought anything like "oh, what does she see in him?!" or "come on, that's so contrived".
I think both Hathaway and Sturgess did a great job of portraying their characters. Emma slowly transformed from a slightly 'spiky' girl with a bit of a self esteem problem to a more confident young woman with life experience. Dexter is a carefree, quite irresponsible young man who gets sucked into the world of TV fame that distances him from the people who love him, and ultimately grinds him down into a convenient relationship. And yet, even as they change (and if anything, diverge from each other) over the years, something remains to draw them back together. Don't get me wrong, this is far less soppy than it sounds - and in fact I thought the film got the balance between 'emotional', funny and serious scenes mostly just right, although a couple felt perhaps too short.
In the end, how much it appeals will probably depend on how much you identify with what happens. For me, several themes struck quite a chord, and I walked out of the cinema feeling that it was definitely a "good film". And that's as much psychoanalysis as is necessary!