Mrthdrb

In the online version of Cambridge Dictionary, there are these definitions for the next two words:

Behaviour = the way that someone behaves

Behavior = a particular way of acting

What would be a difference between behavior and behaviour by some example?

They mean the same thing; behaviour is the British English spelling; behavior is the American spelling. The definitions say the same thing in different ways.

In fact, there is no difference between behaviour and behavior except spelling. The former is preferred in British and Commonwealth English, the latter is the American spelling.

The entries are confusing because there is no single "Cambridge Dictionary." Cambridge University Press actually publishes dozens of different dictionaries. Their website, however, searches them all at once, and returns definitions which may or may not be relevant to you.

When you look up behaviour, you are given entries from the Cambridge Advanced Learner's Dictionary & Thesaurus and the Cambridge Business English Dictionary. When you look up behavior, you are given the entry in the Cambridge Academic Content Dictionary, and from the Cambridge Business English Dictionary a pointer to the entry for behaviour. Because the target audience for each dictionary is different, you see slightly different entries, but this is a quirk of the Cambridge Academic Content Dictionary failing to synonymize the spellings.

I commend you for doing your best with references, but would also recommend you limit searches to a learner's dictionary (e.g. Collins, Oxford, Macmillan, Cambridge, or Merriam-Webster) to avoid this happening in the future.

In short, they're the same word spelt differently in the US and the UK.

It's not very obvious because of the way Cambridge has laid out their pages.

behaviour

noun UK US behavior

behavior

noun [ C/U ] CDN BR behaviour

behavior

noun [ U ]

→ behaviour

Wiktionary's entries are much clearer:

behaviour

Alternative forms

behavior (US)

behavior

Alternative forms

behaviour (UK)

One is the canonical (i.e. so-called "British") spelling and the other is the (idiomatic) American variant.

Note that, in a linguistic sense, there is no such thing as "British" English, anymore than there is "Italian" Italian, or "Spanish" Spanish, "French" French, "Russian" Russian, etc. "British" English represents the canonical form of the language, and American English represents a 'topical' (albeit internationally exposed through modern technology) idiomatic variant, in a similar way that Latin American Spanish is a variant of the canonical form of Spanish that is spoken in Spain, or how "Canadian" French is a variant of the canonical form of French spoken in France, etc.

However, given the large number of users, and particularly the disproportionate use of "American" English in the context of software and the internet, it has become practical to disambiguate between the two in this manner, so I will maintain this artificial distinction below in that understanding. In fact, the use of "US localisation" in software is probably the biggest factor that has lead to the American variant getting broad international exposure. I still remember the time when, English started being separated to US vs UK English in software; I remember feeling rather bemused when I first saw this, but in reality, in the context of selecting the right keyboard, spell checker, currency, date format, units, etc, it made perfect sense. Nowadays though, it is almost seen as a 'competing idiom' to the canonical form (and, as you've seen in the comments in this thread, given the historical relationship that the US has had with Britain, many Americans have rather come to perceive their idiomatic use of the English language and the extent to which it deviates from "British" canonical English as a point of national pride).

In any case, the canonical 'British' spelling (despite the jokes in the comments) is typically the more 'correct' one in a linguistic sense, in that it is typically more faithful to the etymology of the word, preserving the historical links to the words from which they have been derived, underlying phonetic and grammatical rules, etc. This is particularly true with words borrowed from other languages, e.g. Greek and Latin.
'American' spelling, on the other hand, is effectively a more colloquial form, and it is typically more 'phonetic' than grammatical in nature, particularly in the case of loanwords. Consider, e.g. "doughnut vs donut", "organise vs organize", etc. Having said that, note that, occasionally Americans stick to the canonical form too (e.g. "advize" is not a common spelling, despite the typical '-ize' americanisation).

So, if you have a choice, I would go for the so-called 'British' spelling, since this is the canonical form of English. This is particularly true if used in an international setting, where, speaking / writing in "American" English might sound like an odd choice (in the same way that speaking "Indian" English as a non-Indian might sound like an odd choice). On the other hand, when dealing with contexts relating to the US in general, the US variant may be more appropriate; to some Americans, it may even seem 'unpatriotic' or 'conceited' to deviate from the US variant, so, really, it's a judgement call. In formal settings (e.g. academic journal writing), it is usually clear which is preferred from the relevant style manual used.