Bilingualism is the ability and function of dual language acquisition. It is best acquired through consistent exposure in real-life scenarios where grammar can become internalized by the naturally-existing cognitive processes that help us learn new information.
There are two types of bilingualism: early and late. Within the early category there are also subcategories that are proportional to the level of development of the individual. Late bilingualism is basically the learning of an L2 "later" in life.
Sub-categories of Early Bilingualism
Coordinated bilingualism is the type of bilingualism that comes as a result of learning two completely different linguistic systems at the same time. This is a stage of early bilingualism because it is during very early childhood that children are most likely to be exposed to two different language groups by parents who speak different languages and use both at home. It is deemed as "coordinated" because each separate language will be spoken in its entirety and with consistency. As a result, the child will learn both systems, and their subtleties, with exactness and accuracy, and without any problems. It is organized language learning.
During early childhood there can also be compound bilingualism. This type of bilingualism consists on exposure to two different languages in an inconsistent way. This inconsistency occurs when both parents speak a different language plus English AND use them interchangeably; as a result, the child learns the L2 at a more disparate level. Still, the child will get to learn both language systems, but may not be able to appreciate the said subtleties that are ever-present in a consistent language environment.
The second major category of bilingualism is "late" bilingualism. If you are in favor of the "critical period of learning" hypothesis proposed by Penfield and Roberts (1959) and followed by Lenneberg (1967), then you would agree that there is a purported time during our development in which learning can be optimal. Hence, late bilingualism is any L2 learning that happens after that "critical period". That would be the typical case of the modern L2 learner, which often enters the L2 world voluntarily in aims to learn for specific purposes. In this case, the formula remains: consistent exposure in relevant tasks and preferably as close to full immersion as can be.
Bilingualism can be divided in two ways: by the method or circumstances under which a second language is acquired, and the degree to which the second language is perfected. First, a native speaker can learn a second language by relocating—a Vietnamese who moves to the U.S. learns a second language by exposure to English. An American expatriate to France can learn French by exposure to the French culture, etc. (usually augmented by a little formal education as well.) A second way is formal education—I learned German in college as part of my comparative literature degree. (This method of learning a second language can be acquired in grade school or high school, too.) Thirdly, a native speaker can be exposed to the language of a caretaker or companion or a family from two cultures—many American children learn Spanish from their nannies or fellow-workers. A second language can be part of business exposure, such as traveling representatives from U.S. to Japan or China; in this category are any language assimilation needed for or caused by a third motive.
Another way to divide bilingualism is by degree—understanding oral transmission, comprehending written language, informal language use vs. formal language mastery, comprehension with dictionary in hand, tourist language use (“thank you”, “where is the train station?” etc.). Some bilingualism includes formal “correct” grammar usage, while some is simply a lexical comprehension with only a rudimentary understanding of tense, verb forms, etc. A bilingualism that allows translation of literature (the nuances of idiom, figures of speech, vague references, style, etc.--Joseph Conrad, a Polish native, wrote in English) or translation at the U.N. (a famous mis-translation from a Russian-English bilingual translator was “We will outlive you” vs. “We will bury you”) is different from the bilingualism of a landlord collecting rent in a Latino neighborhood.