Hapax legomenon is Greek for "said only once". It is commonly used for words and phrases that appear only once in the history of recorded work. These phrases are often idiomatic, and are frustrating to translate, as they have no other contexts from which to glark their meaning.

My father once wrote a short story called "Hapax Legomenon", which was about a ghost that was cursed to haunt the classics library at Oxford looking for a word that he had come across during his lifetime.

Hapax legomenon, plural hapax legomena, is Greek for "thing(s) said once." A technical term in Biblical studies, classical philology, and related fields, it refers to a word that appears only once in a given literary corpus.

This corpus can be defined in a variety of ways, and the sense in which a historian uses the term "hapax" can usually be deduced from the context.

1. A word can be a hapax within a given book. For instance, the Greek word panoplia, "armour", appears in the gospel of Luke in only one place (11:22), and may therefore be described as a "Lukan hapax." However, the word also appears in Ephesians 6:11 and 6:13, so panoplia is not a hapax in any broader sense. The fact that Luke only uses the word once is interesting for various reasons, but it does not leave us with serious problems of interpretation.
2. It is also possible to talk about a hapax within the oeuvre of a particular author. For instance, the Greek word skemma, "scheme, plot," appears only once in the surviving works of Josephus, but it is found elsewhere in Greek literature (albeit with a slightly different meaning than Josephus seems to intend for it). Again, the fact that Josephus only uses the word once can be significant, but that does not necessarily make it impossible to define.
3. Finally, the broadest sense of the term "hapax legomenon" refers to a word that appears only once in the surviving literature of a given language. One can try to define such a word by working it out from the surrounding context; by guessing at its etymology; by breaking it down into its component parts; by comparing it with similar words in cognate languages; or by consulting commentaries or glosses on the text from the same time period. However, this process is always speculative, and it is further complicated by the fact that we have no way of knowing whether the original author was using a word idiosyncratically, poetically... or even incorrectly!

Sometimes even the final category of hapax can be interpreted without too much difficulty. For instance, the Greek word epidiatassetai, found in Galatians 3:15, is a category-3 hapax, but it is not hard to define: the verb is obviously constructed from the prefix epi-, here meaning "besides, in addition to," and diatassomai, meaning "to appoint, to ordain." It's worth knowing whether Paul made up the word or whether he inherited it from literature now lost to us -- and Bible scholars spill a lot of ink trying to figure out the answer to questions like these -- but in either case, we can be pretty sure what the word means.

Some hapax legomena can be much harder to define than this example, however. The more fragmentary a linguistic corpus, the more trouble we have constructing complete dictionaries, and the more hapax legomena we are forced to reckon with. Technical names for animals, plants, building materials, and so forth, can be especially difficult to define, and sometimes such terms are extremely important for understanding a text. For instance, Leviticus 11 provides a lengthy list of animals which are kosher and animals which are treyf. An observant Jew is obligated to know what God considers appropriate meat to eat, and yet the chapter is filled with hapax legomena whose meaning is unclear. (Today, orthodox traditions err on the side of caution, avoiding all birds besides the ones which are known to be acceptable. This is one application of the process known as placing a fence around the Law.)

There are approximately 1,500 hapax legomena from the third category in the Hebrew Bible, about 400 of which are not directly based on known words. A higher proportion of hapax legomena are found in poetic books like Job and the Song of Songs, though troublesome lists (like the merchandise found in Ezekiel 27) appear in other books too.

Further Reading:
I am not aware of any web pages that discuss New Testament hapax in detail. I'd be glad to hear suggestions.
The following URL leads to a good discussion of hapax in the Hebrew Scriptures, along with some principles of interpretation.
http://www.jewishencyclopedia.com/view.jsp?artid=268&letter=H
I do not know very much about the study of other languages and literatures at all, and encourage other noders to add examples from their own areas of expertise.