In linguistics, the process of change from a pidgin to a creole. Outside linguistics it is also occasionally used, with a rough sense of becoming part of a Creole culture, e.g. Cajun cooking; but in language it has a quite specific meaning. It indicates that instead of being a crude means of communication, a form of language is now a fully-fledged human native language.
A pidgin is a simple language created by attempting communication between adults who don't know each other's language: typical utterances sound comical to us, almost like you fella run along, chop chop, we big fella talkee now. Often but not always it is based on European colonial languages. Pidgins occur anywhere in the world where contact between different peoples necessitates finding or making a common vocabulary with very simple syntax. No-one is a native speaker of a pidgin.
The process of creolization (I have just seen the synonym depidginization) is when this crude working language becomes a language of the cradle. The words creole and cradle are related. Capital-C Creole refers to specific cultures, but lowercase-c creole is a linguist's term for any language that was a pidgin but has acquired native speakers, who learn it in the cradle: typically the children of those who made first contact. The fact that creolization occurs is a testament to some innate property of the human brain that structures things syntactically according to universal principles.
A creole might look to outsiders as if it's still the ludicrously simple shouting-at-servants language, but internally it is now the equal of any other human language. The transition is abrupt and deep. Deaf children with no way of communicating created the Nicaraguan Sign Language spontaneously in a matter of several years. Human language capacity appears to be innate and hard-wired, and everyone uses it unless they're deprived of contact for many years. Children with no 'native' language, only a confusing mix of pidgin, spontaneously convert it to a complete working human language. This is called a creole.
The Bislama creole of Vanuatu interestingly points up that they can be as grammatically complex as any other. Native Vanuatu languages have not only singular and plural but dual and trial. So does Bislama, even though it's based on English, which lacks these. It has inclusive and exclusive 'we' (we-but-not-you versus we-and-you) because its Ni-Vanuatu substrate has these grammatical features, even though English, from which it gets the actual words, does not. For example, yunmitufala 'we (inclusive dual)' comes from the pidgin 'you and me two fellers'. In pidgin it's a silly string of words. In creole it's a complex and sophisticated pronoun that fits neatly in a structure.
At this point see the works and theories of Noam Chomsky, Steven Pinker, and other such, for how the brain handles linguistic structures. I don't feel qualified to comment.
So a creole is a living language that has developed by creolization; the next step is often decreolization, if it's in contact with the donor language. All languages have different registers, including a range based on social prestige of the variety: an acrolect is one highly thought of, and a basilect is one considered low. When you put on your good voice and speak politely you're using an acrolect. In a creole community, speakers are often in contact with and partly know the host or donor language (typically the European colonial language) as well as their own variety. In social contact with Europeans (or whoever) they use the style of creole closest to it, the acrolect. Repeated use of a variety intermediate between creole and donor blurs the difference between them. This is known as the post-creole continuum. For example, there is a continuum between pure Jamaican Creole incomprehensible to outside English-speakers, and merely Jamaican-accented English. It has been, at last partly, decreolized.
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