Also spelled homoioteleuton, and pronounced ho-moyo-TEL-you-ton, this Greek word literally means "similar ending." It refers to a text in which two lines end with the same word or phrase. An English example might look something like this.
Yesterday I went to the mall
wearing only my pajamas.
This seemed to irritate the
people working at the mall
security desk.
Imagine a reader of this simple paragraph being briefly interrupted. Just after finishing the first line, say, she hears a noise and glances up. It is possible that, when she resumes her reading a moment later, her eye might come to rest on the sixth line, rather than the second. In other words, she'll scan the page for the words "the mall" -- the last words she remembers seeing -- and there's a chance she might restart her reading at the wrong instance of the phrase. As a result, she will misread the paragraph as "Yesterday I went to the mall security desk."
Everyone does this sometimes. It could be because the reader is bored, fatigued, or distracted. It could be because the layout of the page in the book is poorly designed or confusing. It could be because the text itself is repetitive, making it easier for a reader to skim or skip lines.
The reason the phenomenon is interesting, and the reason that there is a specific word for it, is that columns of text that end in similar words would sometimes confuse ancient scribes and cause them to mis-copy the texts they were working with.
Let us re-imagine the hypothetical situation I described earlier. Now our reader is faithfully copying the above paragraph out of a book so that she can put it into a letter to a friend. Her eyes are jumping back and forth between the book and her letter, and she makes a mistake: instead of writing the entire thing, she writes only "Yesterday I went to the mall security desk." This kind of error is called parablepsis, literally "looking by the side."
Back in the days before the printing press, when all books were copied by hand, errors of this sort would creep into manuscripts due to fatigue, poor working conditions, or even the sheer incompetence of the scribes. Of course, the line breaks may not be exactly the same in the copy as they were in the original, meaning that the exact nature of the error can be difficult for a contemporary reader to catch.
It is easier to confirm the existence of this sort of scribal error when there are a number of manuscripts to compare with one another. Otherwise, one might simply assume that our author meant to say "Yesterday I went to the mall security desk." In other words, if our reader's letter were to survive a hundred years while the book she was quoting does not, future historians will have no way of knowing she made a mistake (especially if the rest of her writing is terse or awkward). On the other hand, if she were to introduce her letter by saying "Here is a great story that I found in a novel by John Doe," and if that novel by John Doe were to survive, then historians could compare her text with his, and then decide whether the discrepancy comes about due to parablepsis or due to some other factor. (Maybe the omission was deliberate. Maybe our writer was scandalized by the mention of pajamas. History is an art, not a science.)
For an example of an unusual reading arising from homoeoteleuton in the New Testament, see the parablepsis node.