Briefly, a few simplified definitions (see Wikipedia for more complete definitions).

Accent - The pronunciation distinct to a region.

Dialect - The grammar, vocabulary, and pronunciation distinct to a region. Dialects of a language are mutually intelligible. Some dialects have prestige and are used in the mass media and in formal situations. Some dialects are stigmatized.

Language - When dialects of a language are not mutually intelligiblle, linguists classify them as related but distinct languages.

Examples and exceptions:

British English and North American English are dialects. They have differences in grammar, vocabulary, and pronunciation, but are mutually intelligible.

General American (the mass media dialect of American English) and African-American Vernacular English (Black English) are dialects.

Arabic is "one language" for political and cultural reasons. Linguists consider it a family of languages related to old Arabic. The versions are not mutually intelligible. The Egyptian version is widely known due to the popularity of Egyptian mass media, and is sometimes used as a common language when speakers from different regions need to communicate.

The distinction between “language and “dialect” is purely political. For example, Serbian and Croatian are called languages, despite having more intelligibility than accents in different parts of London (and other major cities). Tunisian and Levantine Arabic are considered dialects despite having relatively low mutual intelligibility. Tunisian and Maltese are considered separate languages despite having relatively high mutual intelligibility. “British” and “American” variants would barely qualify as different accents in the context of Arabic or German, or Italian, for that matter (and the variety of accents in UK English is staggering.) See this Wikipedia article if you’re hungry for more information:

One of the things I find most fascinating about languages is watching them grow from other languages and evolve, from dialect to language. And what is the "cutoff line" that divides dialect and language? At what point does a dialect begin to be widely considered it's own language? I am not particularly well-informed on this topic, as it's just a hobby for me, but as far as I know there is no generally agreed-upon answer by experts. I'm sure many, many people on this site are a lot better informed than I, but I do often find myself thinking about language questions like this... As far as African-American Vernacular English (AAVE), I would consider it a dialect, but because languages are constantly evolving, if I'm sure the future has much more in store for it. Cheers!

This really depends on who you ask. There are plenty of arguments for both, however it's usually classified as a dialect. It does in fact have a grammar structure and rules which is the primary argument for it being it's own language, but it's not enough by itself for many to be willing to call it so. Personally, I feel it's close to being it's own language because of those rules and phrases, but it's completely understandable by any English speaker. It has phrases that people will need to ask about at times, but so does any region. Maybe one day it'll be classified as a solid true language soon, but not yet (or maybe it has and I missed the memo).

Phil, so considering all the factors you have stated, is Ebonics/AAVE a language or a dialect?

Irene, yes that's what I meant actually: the rest of the Americans adopting some AAVE terms and expressions.