Learn Hardcore Greek: Αυτή σήμερα είναι λίγο κουρασμένη, αλλά είναι χαρούμενη. - Today she is a little tired, but she is happy.

Αυτή σήμερα είναι λίγο κουρασμένη... explicitly emphasizes “she”. It can sound like “She, today, is a bit tired…”, maybe contrasting her with someone else or with how she is usually.
If you say Σήμερα είναι λίγο κουρασμένη, αλλά είναι χαρούμενη, it’s also correct and more neutral: “Today she is a bit tired, but (she) is happy.”

So yes, you can leave Αυτή out. Keeping it adds emphasis or contrast on “she”.

Does Αυτή here mean “she” or “this (woman)” in English?

Αυτή can mean both “she” and “this (woman)”, depending on context.

As a personal pronoun, it means “she”.
As a demonstrative, it can mean “this one (feminine)”.

In this sentence, with no noun after it and in a simple context like “She is a bit tired today…”, it is most naturally understood as “she”.
If it were clearly demonstrative, you’d often see a noun:

Αυτή η γυναίκα είναι κουρασμένη. – This woman is tired.

So here, read it as “she”.

Why is the word order “Αυτή σήμερα είναι…” and not “Σήμερα είναι αυτή…” or “Σήμερα είναι λίγο κουρασμένη αυτή”?

Greek word order is fairly flexible, and all of these can be grammatical, but the focus changes slightly:

Αυτή σήμερα είναι λίγο κουρασμένη…
Focuses first on “she” – She, today, is a bit tired… (maybe compared to others or to other days).
Σήμερα είναι λίγο κουρασμένη, αλλά είναι χαρούμενη.
Very natural, neutral order: “Today she is a bit tired…”
Σήμερα αυτή είναι λίγο κουρασμένη…
Adds contrast on αυτή: Today *she is a bit tired* (as opposed to someone else).
Σήμερα είναι λίγο κουρασμένη αυτή.
Possible but more marked; αυτή at the end can sound like you’re contrasting or clarifying the subject: Today it’s her who is a bit tired.

The original order highlights “she” and then “today”.

Why do κουρασμένη and χαρούμενη both end in -η?

Adjectives in Greek agree with the noun (or pronoun) in gender, number, and case.

The subject is Αυτή → feminine, singular, nominative.
So the adjectives also must be feminine, singular, nominative:
- κουρασμέν–η
- χαρούμεν–η

Compare with other genders:

Masculine:
- Αυτός είναι κουρασμένος, αλλά είναι χαρούμενος.
Neuter:
- Αυτό είναι κουρασμένο, αλλά είναι χαρούμενο.

The -η endings show feminine singular here.

What exactly does λίγο do here, and where can it go in the sentence?

Λίγο means “a little / a bit / slightly” and modifies the adjective κουρασμένη.

είναι λίγο κουρασμένη = she is a bit tired (not very tired).

Common positions:

Είναι λίγο κουρασμένη σήμερα. – Very natural.
Σήμερα είναι λίγο κουρασμένη. – Also very natural.
Είναι κουρασμένη λίγο σήμερα. – Understandable but less natural; λίγο usually goes before the adjective.

So λίγο is an adverb here, softening the adjective.

Why is there no article, like η, before κουρασμένη or χαρούμενη?

After the verb “to be” (είμαι), when you describe someone’s state or quality, Greek normally uses the adjective without the definite article:

Είναι κουρασμένη. – She is tired.
Είμαι χαρούμενος. – I am happy.

Using the article would usually be wrong or change the meaning:

Είναι η κουρασμένη. – Sounds like “She is the tired one” (labeling or identifying her in a group).

In your sentence it’s a simple description, so: no article.

Why is είναι repeated? Can we say «…είναι λίγο κουρασμένη, αλλά χαρούμενη» instead?

Both options are possible, with a small nuance difference:

…είναι λίγο κουρασμένη, αλλά είναι χαρούμενη.
- Repeats είναι → clear, balanced, slightly more emphatic.
…είναι λίγο κουρασμένη, αλλά χαρούμενη.
- Omits the second είναι, which is understood from context.
- Sounds a bit more compact and colloquial.

Grammar-wise, both are correct. Repeating είναι is very natural and especially common in careful or written Greek.

What does αλλά mean here, and do we always put a comma before it?

Αλλά means “but”, introducing a contrast:

…είναι λίγο κουρασμένη, αλλά είναι χαρούμενη.
– She is a bit tired, but she is happy.

About the comma:

When αλλά connects two full clauses (each with its own subject + verb), a comma before αλλά is standard:
- …, αλλά είναι χαρούμενη.
If it connects two short phrases inside one clause, there might not be a comma:
- Είναι κουρασμένη αλλά χαρούμενη. (no second είναι, no comma necessary)

So, in your original sentence, the comma before αλλά is correct and normal.

How would the sentence change if the subject were a man instead of a woman?

You would change the pronoun and the adjective endings to masculine:

Αυτός σήμερα είναι λίγο κουρασμένος, αλλά είναι χαρούμενος.
- Αυτός = he
- κουρασμένος = tired (masculine)
- χαρούμενος = happy (masculine)

Same structure, just masculine forms instead of feminine.

Could this sentence be in the past tense? How would I say “She was a bit tired today, but she was happy”?

Yes. You would change είναι (present) to ήταν (past):

Αυτή σήμερα ήταν λίγο κουρασμένη, αλλά ήταν χαρούμενη.
- ήταν = “was” (for all persons singular & plural)

Everything else stays the same because the adjectives describe her state; they don’t change with tense, only with gender/number/case.

Is σήμερα always placed near the verb, or can it move around?

Σήμερα (today) is an adverb of time and is quite flexible in position. All of these are natural:

Σήμερα είναι λίγο κουρασμένη, αλλά είναι χαρούμενη.
Είναι λίγο κουρασμένη σήμερα, αλλά είναι χαρούμενη.
Αυτή σήμερα είναι λίγο κουρασμένη, αλλά είναι χαρούμενη.

The main rules:

It usually goes somewhere near the verb or at the start of the sentence.
Moving it mostly changes emphasis, not grammatical correctness.