Mat 8:12 But the children of the kingdom shall be cast out into outer darkness: there shall be weeping and gnashing of teeth.
However, the children of the realm are going to be tossed out into the darkness, the outside, the known world. It is going to be the crying and the biting of the teeth.
This verse uses a number of phrases associated with Christ. Christ uses the phrase "children of the kingdom" here to refer to the children of Israel, but elsewhere, he uses it to refer to the children of the "realm of the skies" or to the "good seed" (Mat 13:38). this is the first time the phrase "outer darkness". It is used elsewhere in parables (Mat 22:13, Mat 25:30). It is also the first time the phrase "weeping and gnashing of teeth" is used. Christ's use of the ending phrase of this verse is typical of Christ's use of exaggeration and odd phrases as humor. For more on this topic, see this article.
The Greek word translated as "but" joins phrases in an adversarial way. Since it always falls in the second position, translating it as "however" often captures its feeling better.
The word translated as "the children" more generally means "child" and more specifically means "son." It is plural.
The word translated as "kingdom" can be the region, the reign, the castle or the authority of a ruler. Christ does not seem to use it to mean a physical region, so its translation as "reign" or "realm" seems more appropriate. This is especially true because the "reign" of a king means the execution of his will.
"Cast out" is a Greek verb that means "throw out." Depending on the context, it can mean "toss out", "turn out," or "take out." It is usually translated as "cast out" in the NT. The general idea of is "to throw without caring where something falls," so it isn't like putting something into a specific place. The verb is future passive. Christ uses it most commonly to refer to "tossing out" either "demons" or people's lives into the trash.
The word translated as "darkness" means "darkness" and "gloom" and it is a metaphor in Greek for ignorance and the nether world. It is introduced with an article "the."
The word translated as "outer" is the adjective that means "more outside," but it is preceded by an article, so it acts as a noun, "the outside".
The word translated as "there" means "there" or "in that place" but it also means "the intelligible world," that is, the world we understand. It refers to a place much more strongly than our word "there" which can be a simple helper to introduce a verb of being.
When the verb "to be" appears in the future and so it is translated as "shall be," but in Greek, it doesn't require the word "there" to mean something like "there will be."
The "weeping" come from a noun form of the verb "weep" which means "to weep", "to cry", "to lament," and "to wail."
The Greek word translated as "and" is used as the conjunction "and", but it also is used to add emphasis ("also") and, in a series, is best translated as "not only...but also."
The word translated as "gnashing" which primarily means "biting." However, it also means "chattering". Used with the word "darkness," it gives a sense of extreme cold, but the same phrase is also used to refer to tossing people in the fire (Mat 13:50).
The word translated as "teeth" means tooth but it is a metaphor for the pain of grief. Toothaches are serious, continued, painful conditions.
The "gnashing of teeth" phrase captures both the sense of what we call "back biting" among people and the pain of loss.
οἱ (article) "The" is the Greek article, "the," which usually proceeds a noun and, without a noun, takes the meaning of "the one" or, in the plural, "the ones." Here it is separated from its noun by a conjunction.
δὲ (partic) "But" is from de which means "but" and "on the other hand." It is the particle that joins sentences in an adversarial way but can also be a weak connective ("and") and explanation of cause ("so") and a condition ("if").
υἱοὶ (noun pl masc nom ) "Children" is from huios, which means a "son," and more generally, a "child." It is used generally to refer to any male descendant and, more generally, any descendent of either sex.
ἐκβληθήσονται (3rd pl fut ind pass) "Cast out" is from ekballo and means "throw out", "cast out of a place,"and "expose." Ek means "out of", "from," and "away from." Ballo is "to throw" or "to scatter."
εἰς (prep) "Into" is from eis, which means "into (of place)," "up to (of time)", "until (of time)", "as much as (of measure or limit)", "as far as (of measure or limit)", "towards (to express relation)", "in regard to (to express relation)", "of an end or limit," and "for (of purpose or object)."
καὶ (conj) "And" is from kai, which is the conjunction joining phrases and clauses, "and," or "but." After words implying sameness, "as" (the same opinion as you). Used in series, joins positive with negative "Not only...but also." Also used to give emphasis, "even", "also," and "just."