In the context of translation, there are 2 kinds of certification. My clients are often told they need a "certified translation" or a translation done by a "certified translator". Unfortunately, even the authorities that require certification usually don't know what it means! [fontawesome icon="bookmark" circle="yes" size="medium" iconcolor="" circlecolor="#6E5C46" circlebordercolor="#6E5C46"]
When someone says that your translation must be done by a "certified translator", they mean the translator must have an official license of some kind.
For example, for a translation to be filed with the county recorder or in court, the translator must be a California Certified Court Interpreter or a translator certified by the American Translators Association. I have both of these certifications, see my qualifications for details.
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When someone says that your foreign language documents must be accompanied by a "certified translation", they mean that the translation must have a declaration attached. The declaration (or affidavit) is a sworn statement signed by the translator stating their qualifications and that the document was translated to the best of their ability.
Note that when we talk about the translation being "notarized", that means having the translator sign their certification (declaration) in front of a notary public. See the FAQ about notarization for details.