Luke 19:10 For the Son of man is come to seek and to save that which was lost.
Because he shows up, this son of the man, to search for and rescue the thing having been destroyed.
This is another reference to information that is not in Jesus's statements. The form of "that which was lost" is neutral and doesn't match anything in the previous verse, the "house" or the "son", both of which were masculine. The "for" seems to indicate an answer to a question.
The word translated as "for" introduces a reason or explanation so "because" and, in questions, "why." However, since this word always appears in the second position, it is more like an aside remark like, "consequently" or "as a cause".
The phrase "the son of man" is the common way Christ refers to himself. It is discussed in detail in this article. Its sense is "the child of the man" or "this son of that man". The word translated as "son" more generally means "child" or "descendant". The Greek word for "of man" in the singular means "person" and "humanity" and "people" and "peoples" in the plural.
The word translated as "is come" primarily means "to start out" but Christ usually uses it to mean "come" but not always. It indicates movement, especially its beginning, without indicating a direction toward or away from anything, so it works either as "come" or "go," but it is more like our phrase "being underway." Our English word "show up" captures both the "start" and "come" ideas.
The Greek verb translated as "to seek" has a variety of meanings around the idea of "searching" and "desiring". It has a sense of seeking with a specific aim.
The Greek word translated as "and" is used as the conjunction "and", but it also is used to add emphasis ("also").
"To save" is the Greek word that means "to keep alive" when applied to people or "to keep safe" when applied to things. Christ uses it to mean "rescue" in most cases.
The word translated as "that which" is the Greek definite article, which usually precedes a noun and, without a noun, takes the meaning of "the one." The Greek article is much closer to our demonstrative pronouns ("this", "that", "these", "those"). See this article for more. The form is singular neutral, "this thing".
The word translated as "was lost" means primarily to destroy or demolish, and "lost" in the sense of destroyed. It is an adjective in the perfect then, "having been destroyed"
ἦλθεν ( verb 3rd sg aor ind act ) "Is come" is erchomai, which means "to start," "to set out", "to come", "to go," and any kind of motion. It means both "to go" on a journey and "to arrive" at a place.
γὰρ (partic) "For" comes from gar which is the introduction of a clause explaining a reason or explanation: "for", "since," and "as." In an abrupt question, it means "why" and "what."
ὁ υἱὸς (noun sg masc nom) "The Son" is huios, which means a "son," and more generally, a "child." It is used generally to refer to any male descendant.
τοῦ ἀνθρώπου (noun sg masc gen) "Of man" is anthropos, which is "man," and, in plural, "mankind." It also means "humanity" and that which is human and opposed to that which is animal or inanimate.
ζητῆσαι ( verb aor inf act ) "Seek" is zeteo, which means "inquire for", "search for", "seek after", "desire", and "feel the want of."
καὶ (conj/adv) "And" is 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."
σῶσαι ( verb aor inf act ) "Save" is sozo (soizo), which means "save from death", "keep alive", "keep safe", "preserve", "maintain", "keep in mind", "carry off safely," and "rescue."
τὸ (article sg neut acc) "that which" is the Greek definite article, which usually precedes a noun and, without a noun, takes the meaning of "the one" or, in the plural, "the ones."
ἀπολωλός. (part sg perf act neut acc) "was lost" is apollymi, which means "to demolish", "to lay waste", "to lose", "to perish", "to die", "to cease to exist," and "to be undone."