Learn Hardcore Latin: Puer quietus domi manet. - The quiet boy stays at home.

manet ends in -t, which marks 3rd person singular in the present tense (he/she/it stays).
The word that matches this in number (singular) and is in the nominative case is the subject.

puer is nominative singular (from the dictionary form puer, pueri, “boy”), so puer is the subject, and manet is the verb:

puer = subject (“the boy” / “a boy”)
manet = verb (“stays”, “remains”)

What exactly is quietus doing in this sentence?

quietus is an adjective meaning “quiet, calm, at rest,” and it is in the nominative masculine singular, agreeing with puer.

Syntactically, it can be understood in two closely related ways:

Attributive adjective: “the quiet boy stays at home.”
- Here it just describes which boy we’re talking about.
Predicate adjective with a “linking-type” verb: “the boy stays (i.e. remains) quiet at home.”
- Here quietus is more like a result/state of staying: he is quiet while he stays.

Because manere (“to remain, to stay”) often behaves like a linking verb (similar to “to be, become”), many teachers will prefer to explain it as:
puer (subject) – manet (linking verb) – (predicate adjective).