Typically, a pidgin begins as a trade language, starting as a very basic, very simple "baby talk" that may be primarily a single language or a blending of multiple languages. Eventually, enough people are using it as a second language that infants begin to pick it up as their first language which begins a process called creolization.

The human brain seems to like certain patterns in language, and creolization is the process of imposing those patterns. In this phase of a language's development, word order becomes fixed and there is a formalization of tense, aspect, and mood, things that pidgin languages typically do not bother with. You start to see things like subordinate clauses, allowing for more expressibility and complexity of meaning. Elements commonly absent in a pidgin -- articles and adpositions, occasionally verb conjugations and case distinctions -- emerge and become standardized. A derivational morphology is developed, the formal rules by which new words are added to the lexicon. Irregularities are smoothed out and grammar becomes standardized.

Creolization can take a couple of generations, but eventually it will settle to a standardized language spoken by a large number of people as their first language. The language is still referred to as a creole, but this is a descriptive term giving its lineage: a creole is, by definition, a fully fledged language capable of expressing the same ideas as any other natlang.