African-American Vernacular English, also known as Black English, is a dialect of American English, which is itself a dialect of English. There is a formal written English which none of us speaks; all of us speak a dialect. There is a standard American English that tends to be closer to the formal written version, but is still not entirely the same. Even after becoming fluent in the formal written style, speakers tend to persist in speaking according to their original dialect.
The line is not always clear between what is an accent and what is a dialect. Some would claim that there are several Southern dialects, others that they are merely accents. Scottish English*, Pennsylvania Dutch English**, and Indian English***, and Newfoundland English are generally regarded as fully dialects, not merely accents, as they are different in structure. There are more.
AAVE is believed to have originally sprung from various British dialects - compare pirate talk "Whar ye' be goin?" - and subsequently influenced by Southern American speech, which was also a product of specific British dialects, usually western and Irish. The "be" verbs in AAVE are quite complicated and nuanced, actually. While all non-majority dialects in all languages are looked down upon and regarded as "just wrong," we look down on AAVE more than the others. This may be changing, as AAVE is influencing not only slang but the standard American spoken dialect through entertainment media. It is not only used ironically or for effect among the young. It is just blending in. The difference between influencing the majority culture and having your culture appropriated is largely one of perspective.
There is little or no African in Black English. That myth keeps resurfacing, but there isn't any linguistic support for it. It has also become more standardised since the middle of the last century. Slaves certainly did not have much contact with other African-Americans, slave or free, in other parts of the country. Their speech was related by history, but not always fully mutually intelligible. The great migrations north in the 1910's and 1930's and 40's created a more agreed-upon version in the cities, and radio, recordings, movies, and television spread this throughout the country.
We have looked down on it because of its associations, not because it is objectively worse or less standard than listening to a Newfie. It was spoken by people who were poor and had less education, so we came to regard it as more substandard than we might if it were simply a regional dialect.
Every schoolchild should learn to write formally. While formal writing can sometimes admit of spoken dialect for effect, as with Mark Twain, this is kept to a minimum. In speech, it is advantageous to be able to use a bland middle-American speech at need. Even prestigious accents are not welcome in all situations. People who read a great deal find that their spoken and formal written expression bleed into each other more than those who do not. I would be at an extreme of that, as my writing relies a great deal on my spoken expression, while in speaking I use written forms more than most people do. I am no longer sure whether this is ironic and for humorous effect or just the way I talk. People who are familiar with both my communication styles say it is easy to hear my voice in my writing.
*Not to be confused with Scots Gaelic
**Not to be confused with Pennsylvania Dutch, which is a dialect of German,
***As in India, not Native Americans.