Therefore, since God was fond of the world order, he appointed the son unique in order that all believing in him might not want to cease to exist but might possess life eternal.
Jhn 3:16 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
Interesting and Hidden Aspects:
The term translated as "only-begotten" means "one of a kin" but more generally it means "unique" and "singular." In grammar, it also means a word having the same form in all genders, and it is kind of funny that the word, "monogenes," is itself a monogenic word, having the same form in all genders.
Using this word, Christ is clearly implying that he is in some way the only son of God, the same blood and kin. He is also saying that he is "unique", which fits with the context here. He has said that he is unique in having been raised up to heaven and returning, which is the larger context of the conversation.
In this verse, we see the phrase "shall not perish," which was added to the previous verse in the KJV and Latin Vulgate.
The negative "not" used here is the Greek negative of a subjective opinion, commands, and requests. The sense is that "you don't want" to do something, not that it isn't done or don't think something that might be true. If it wasn't done or wasn't true, the objective negative of fact would be used. The sense here is that the person doesn't "want" to cease to exist.
The word translated as "shall...perish" means "to cease to exist." The form is one of possibility not the future tense. With the negative used, the sense is "might not want to cease to exist". This seems in indicate that those who do not trust in him do not want to continue to exist. The sense is that not trusting in his promise equates to no trusting in a continued life and having access to it.
"Everlasting" is an adjective based on the word that means "age" or "eon." It has the sense of "perpetual" or "ageless."
The word translated as "life" means "living" but it also means "substance", "existence," and "property." Christ uses it to mean "existence" beyond physical life. See this article on the various Greek words that Jesus uses to describe various aspects of life.
Christ does not seem to be suggesting that they are punished in hell, as in the current vision of the afterlife, but simple that they will be separated ("judged") and not continue in the next age of life.
ἠγάπησεν (3rd sg aor ind act) "Loved" is from agapaô (agapao), which means "to be fond of", "to greet with affection", "to persuade", "to caress", "to prize", "to desire", "to be pleased with," and "to be contended with." This love is more associated with affection than passion.
τὸν μονογενῆ "Only begotten" is from monogenes, which means "the only member of a kin", "only", "single", "unique", "one and the same blood," [in grammar] "having the same form in all genders," and [of plants] "growing only in in one place."
ὁ πιστεύων (part sg pres act masc nom) "Believeth" is from pisteuô (pisteuo), which means "to trust, put faith in, or rely on a person", "to believe in someone's words", "to comply", "to feel confident in a thing," and "to entrust in a thing."
εἰς "In" is from eis (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)."
αὐτὸν "Him" is from autos (autos), which means "the same," and the reflexive pronouns, "myself", "yourself", "himself", "herself", "itself," or the oblique case of the pronouns, "him", "her," and "it." It also means "one's true self," that is, "the soul" as opposed to the body and "of one's own accord."
μὴ "Not" is from mê (me), which is the negative used in prohibitions and expressions of doubt meaning "not" and "no." As οὐ (ou) negates fact and statement; μή rejects, οὐ denies; μή is relative, οὐ absolute; μή subjective, οὐ objective.
ἔχῃ (3rd sg pres subj act) "Have" is from echô (echo), which means "to have", "to hold", "to possess", "to keep", "to have charge of", "to maintain", "to hold fast", "to bear", "to keep close", "to keep safe," and "to have means to do."
ζωὴν "Life" is from zôê (zoe), which means "living", "substance", "property", "existence," and, incidentally, "the scum on milk." It has the sense of how we say "make a living" to mean property. Homer used it more to mean the opposite of death.