Something that is credible is capable of being believed. The word is synonymous with plausible and believable, and it can apply to both things and people. For example, you might say that a report on the benefits of a new drug is credible if it is researched and well sourced, and you might also call the author credible if he or she is an expert in the field. Credulous, which applies only to people, means willing to believe. It’s synonymous with gullible and, like that word, usually has negative connotations. Someone who is credulous is too willing to believe and hence easily tricked or taken advantage of. The negative connotations have not always been there in historical use of the word, but they are unavoidable in modern usage.
The words, along with their corresponding nouns credibility and credulity, are easily mixed up due to their shared first syllable. Just remember that credible is the antonym of incredible, a word that all English speakers should be comfortable with. The antonym of credulous, incredulous, is not as common, and it is similarly often confused with incredible, but if you can manage to keep credible and incredible straight, then the other two should be easy enough.
History
Though the words came to English by different paths and at different times, they share the Latin root credere, meaning to believe, and they both come almost directly from their Latin equivalents (with influence from French). Credible comes from credibilis, meaning worthy of being believed,1 and credulous comes from crēdul-us.2 The earliest instance of credible listed in the Oxford English Dictionary is from Chaucer (late 14th century), and credulous first appears toward the end of the 16th century. When Shakespeare used credulous in Othello, written at the start of the 17th century, it already had the negative connotations it bears today.
Examples
Americans would rather have a credulous believer of supernatural weirdness, than a skeptic. [Chicago Tribune]
These beliefs however are not backed up by credible, scientifically verifiable proof that aliens have visited here or that they actually exist at all. [Irish Times]
And there will always be credulous Western reporters who will take Mahmoud Ahmadinejad’s word that Iran’s intentions are peaceful. [Wall Street Journal]
Contrary to the claims in The Sunday Age, there is no credible evidence that it is not a humane method of slaughter and considerable evidence that it is. [Sydney Morning Herald]
Action-thrillers tend to push the boundaries of credibility. [National Post]
There was a persuasive range of experts, with only one instance of that notorious strain on the viewer’s credulity, the American professor with the joke comb-over. [Telegraph]
Sources
1. Credible in the OED (subscription required) ↩
2. Credulous in the OED ↩