Learn Hardcore Korean: eoje nalssiga chuwosseoyo. - Yesterday’s weather was cold.

The particle 가 is the subject marker. It tells you that 날씨 (“weather”) is the subject of the sentence—i.e. the thing that is doing or experiencing the action/state (being cold).

Why is the verb ending -었어요 used in 추웠어요 instead of -아요 or -아요?

추웠어요 comes from the adjective (descriptive verb) 춥다 (“to be cold”). To make it past tense polite, you attach -었어요. Because 춥- ends in the vowel 우, you use -었- rather than -았- (vowel harmony rule).

Can I say 날씨가 어제 추웠어요 instead of 어제 날씨가 추웠어요?

Yes. Korean has a relatively free word order as long as you keep the verb at the end. Placing 어제 later emphasizes 어제, but both orders are grammatically correct:
• 어제 날씨가 추웠어요
• 날씨가 어제 추웠어요

What part of speech is 추웠어요? Is it a verb?

In Korean, descriptive words like 춥다 are called 형용사 (adjectives/descriptive verbs). They behave like verbs: they take tense endings (past, present, future) and speech-level endings (polite, formal, casual). So 추웠어요 is the past polite form of the adjective 춥다.

Why don’t we need a copula 이다 in this sentence?

Because 춥다 itself means “to be cold.” When you use an adjective/descriptive verb as the predicate, you don’t need 이다. That copula is only for nouns or noun-based predicates (e.g. 학생이다 “to be a student”).

What does 어제 do in the sentence, and can I move it around?

어제 is an adverb of time meaning “yesterday.” Time adverbs typically appear at the beginning or anywhere before the verb. Moving it changes emphasis but not the basic meaning:
• 어제 날씨가 추웠어요 (neutral)
• 날씨가 어제 추웠어요 (emphasizing “yesterday”)
• 추웠어요 어제 날씨가 (unnatural—verb must stay at the end)

Is 추웠어요 formal or casual speech?

The ending -어요 is polite casual speech (also called 요-form). It’s appropriate for strangers or people you’re not very close with. More formal would be 추웠습니다; more casual (with close friends) would be 추웠어.

Can I drop 날씨가 and just say 어제 추웠어요?

Yes. In context, you can omit the subject if it’s clear what you’re talking about. 어제 추웠어요 would mean “It was cold yesterday,” leaving 날씨가 implicit.