Over the years I've read quite a few alien stories. Heck, I've even written a couple myself. As I see it, the easiest alien story is one where all the characters and the location are alien. In other words, everything is "not us". In that case, you don't have to decide what the characters know--or don't know (or understand). They'll all be familiar with the same references. Kama Sutra Lovers is this type of story. The book's setting is Elyria and all the characters--at least all the major characters--are natives of Elyria.
In the second type alien story, one of the characters is not native to an alien setting. In this story, the author creates the setting and then observes it through the eyes of the outsider. The Mystic Valley books are stories of this type. Everything is not only unfamiliar to the hero/heroine, but the reader. In many ways, this is the most difficult type alien story to write because the author must convey a strange world, customs, culture while maintaining the personality of the hero/heroine. What will they know/understand and how do they find out about the new world? Some authors may even be tempted to dump information rather than let their character discover that world on their own.
In the third scenario, the alien is on Earth. This presents the problem of determining what the alien knows about Earth and how quickly he/she assimilates. The world is familiar to the reader so the author doesn't have to explain the setting for the reader's sake. However, the author must still be careful when viewing the surroundings through the outsider's eyes.
In the second and third scenarios I also include time travel novels. In Linda Howard's Killing Time, she beautifully addresses this issue when her FBI agent from the future (though well prepared) is continually snagged by idiomatic language differences.
On one hand, it's too simplistic to assume the alien character will understand everything immediately. On the other hand, the story will screech to a halt if the author stops to explain every itty bitty little thing as they go along. I like to assume reasonable intelligence in my aliens and stop occasionally to show them figuring out/asking about a particularly puzzling issue.
The picture at the top of the post is a perfect example of the type of pitfall I think would be interesting, even amusing to have in a story. Perhaps the alien in this case has numerous little horns to hide. What to do? A normal hat wouldn't fit over his head... Or perhaps he's a dragon and must hide his crest. Hmmmm.
What's your favorite alien story?