This happens to be my main field of research, so I hope you’ll permit me to provide a very detailed (but interesting, I hope) explanation.
How to handle puns and wordplay is one of the most extensively studied problems in the field of translation; there have been entire books written on the subject. Translation theorists have categorized the possible strategies as follows:
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Pun to pun. In this strategy, you find a pun in the target language that (ideally) matches the meaning of the pun in the source language, or if this is not possible, you find a new pun in the target language that still makes sense in the context of the document you’re translating. Needless to say, this strategy can be extremely difficult to apply, and in many cases it’s simply not possible to use at all. An example of a successful application of this strategy can be found in the film Top Secret: in the original English version, the main characters encounter a coughing horse and ask its driver if he’s OK. The diver replies, “Oh, he caught a cold the other day and he’s just a little hoarse.” One subtitler rendered the driver’s reply instead that the horse’s voice was just a little “rostig”, meaning “rusty” in English. But in German one of the words for “horse” is Ross, so pronouncing “rostig” with a slightly overstressed first syllable makes it easy for German speakers to spot the pun. Now, of course, “rusty” isn’t exactly the same meaning as “hoarse”, but it’s close enough that the joke makes sense in the context of the scene.
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Pun to conventional language. In this strategy, you simply ignore the double meaning of the pun and translate only one of the meanings. The humour is thereby lost.
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Pun to non-punning wordplay. In this strategy, the translator substitutes some other type of wordplay for the pun. This can be, for example, rhyming, alliteration, or a malapropism. So in this case, the double meaning may be lost, but the whimsical nature of the text is at least preserved.
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Omit the language containing the pun. Here, the translator excises not only the pun but also the surrounding context. That is, the sentence or paragraph containing the pun is simply omitted from the target document. Needless to say, this strategy also loses the humour. It is generally used only as a last resort, such as when a work of literature includes a very long string of puns that are so language- and culture-specific that they couldn’t possibly be translated using a humour-preserving strategy, and rendering them without the double meanings would be nonsensical.
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Leave the pun in the source language. This strategy can work if you’re reasonably sure that your readership/audience is familiar enough with the source language to get the joke.
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Introduce a new pun elsewhere. In this strategy, you would normally adopt strategy #2 for the pun in question, but then make sure to add a new target-language pun somewhere else in the document. This way you preserve the quantity of humour in the document, even if it may appear in a different place in the target document.
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Introduce new material containing a pun. This strategy is essentially the opposite of strategy #4; you add some entirely new material into the target document that contains a pun. Again, this strategy preserves the amount of humour, but not its location. But it’s definitely not feasible for certain types of documents, such as films: no director is going to shoot new scenes for the foreign release just because the translator handling the dubbing or subtitling came up with some great new jokes. But when translating something like a novel, this strategy might be available to the translator.
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Editorialize. This strategy involves explaining the source-language pun, using something like a footnote or endnote where the translator directly addresses the reader. Again, this is something that doesn’t work in dubbing or subtitling but might be appropriate for printed literature.
The first strategy might be viewed as the “holy grail” of successful translation, though in practice almost never works, and translators fall back to some other strategy. The main factors determining whether you can translate a pun with a pun are how related the source and target languages are, how deep a knowledge of the source and target languages you have, what sorts of resources you have available to you (dictionaries, thesauri, etc.), and how good you are at coming up with puns.
My work involves developing digital translation tools to help translators faithfully translate puns (i.e., using strategy #1). Computers can already locate puns and identify their meanings with a high degree of accuracy, so the next step is to get them to propose translations that can be checked and modified by a human translator. We’ve actually built a prototype system for this and will be testing it with real-life translators in a user study next month. If you’re interested in learning more, you can check out the website for the research project, Computational Pun-derstanding.