Learn Hardcore Hungarian: Ő ma jól alszik.

want to emphasize the person:
Ő ma jól alszik. – It is *he/she who is sleeping well today (as opposed to others).*
need to make clear who you’re talking about in a longer context (several people in the discourse).
contrast with another person:
Ő ma jól alszik, de Péter nem. – He/She is sleeping well today, but Peter isn’t.

If there is no special contrast or emphasis, Ma jól alszik. is often more natural in everyday speech.

Does Ő mean “he” or “she”?

Ő can mean he, she, or even it (for a personified thing like a pet), because Hungarian pronouns do not mark gender.

So Ő ma jól alszik. can be translated as:

He is sleeping well today.
She is sleeping well today.

Only context tells you which one is meant.

What tense is alszik? Why is it translated as “is sleeping” rather than “sleeps”?

Alszik is in the present tense, indefinite conjugation.

Hungarian has only one present tense, and it covers both:

English simple present: He sleeps well.
English present continuous: He is sleeping well.

The meaning is disambiguated by context and time adverbs:

Ő jól alszik. – He/She sleeps well (in general).
Ő ma jól alszik. – Today he/she is sleeping well (today, on this occasion).

So Ő ma jól alszik. is most naturally understood as He/She is sleeping well today.

Why is the verb alszik and not something like alud? What is the dictionary form of this verb?

The dictionary form (infinitive) of the verb is aludni – to sleep.

Alszik is the 3rd person singular present tense form of that verb. Many Hungarian verbs change their stem when conjugated. With aludni, you often see alsz- as the present-tense stem:

(én) alszom / alszok – I sleep / am sleeping
(te) alszol – you sleep
(ő) alszik – he/she sleeps
(mi) alszunk – we sleep
(ti) alszotok – you (pl.) sleep
(ők) alszanak – they sleep

So:
aludni (to sleep) → alszik (he/she sleeps).

What kind of verb is alszik? I heard something about “-ik verbs” in Hungarian.

Yes, alszik is an -ik verb, meaning the 3rd person singular form ends in -ik.

Traditionally, -ik verbs had a slightly different 1st person singular ending (-om/-em/-öm), but in modern colloquial Hungarian both versions are widely used:

alszom (more standard/formal)
alszok (very common in speech)

Both are accepted; alszom is more “bookish”, alszok is everyday speech. The rest of the forms behave like normal verbs.

What is the role of ma in the sentence, and why is it placed before jól alszik?

Ma means “today”, and it is a time adverb.

In neutral Hungarian word order, elements often appear in a roughly Topic – (Focus) – Verb – Rest structure, and time adverbs usually come early in the sentence.

Here:

Ő – topic (who we are talking about)
ma – time (when)
jól – manner (how)
alszik – verb

So Ő ma jól alszik. literally orders the information as:
As for him/her – today – well – sleeps.

Moving ma can change emphasis slightly:

Ma ő jól alszik. – Today it is he/she who is sleeping well (emphasis on *today and contrast with other days/people).*
Ő jól alszik ma. – still grammatical; often sounds more “afterthought-like”: He/she sleeps well today, with ma less emphasized.

The original sentence Ő ma jól alszik. is a neutral, natural order.

What is jól exactly? Why not just jó?

Jó is an adjective: good.
Jól is the corresponding adverb: well.

In English:

good (adj) → well (adv)
He is a good sleeper. / He sleeps well.

In Hungarian:

jó (adj) → jól (adv)
Ő jó. – He/She is good.
Ő jól alszik. – He/She sleeps well.

So jól answers the question “How does he/she sleep?”

Why is there no article in the sentence? In English we say “He sleeps well today,” with he as a pronoun, but Hungarian has no “the” or “a” anywhere.

Pronouns in Hungarian never take articles, just like in English you don’t say “the he” or “a he”.

So Ő stands alone as the subject.

If the subject were a noun, you might see an article depending on whether it is definite:

A fiú ma jól alszik. – The boy sleeps well today.
Fiú ma jól alszik. – rarely used; would sound like “A boy sleeps well today” in some special context.

In Ő ma jól alszik., we only have a pronoun subject, so there is naturally no article.

How would the sentence change if I want to say “They are sleeping well today”?

You need to change both the pronoun and the verb to plural:

Ők ma jól alszanak. – They are sleeping well today.

Breakdown:

Ők – they (plural of ő)
alszanak – 3rd person plural present of aludni (stem: alsz-
- plural ending -anak)

So:

Ő ma jól alszik. – He/She sleeps well today.
Ők ma jól alszanak. – They sleep well today.

Could the sentence also mean “He/She will sleep well today”? Hungarian often uses the present for the future, right?

Yes, Hungarian often uses the present tense with a time adverb to refer to the future, so context matters.

However, with ma (“today”) and alszik (sleep), Ő ma jól alszik. will almost always be understood as:

He/She is sleeping well today
or
He/She sleeps well today (on this specific day).

For a clear future meaning, Hungarians typically use holnap (tomorrow), ma este (this evening), etc., or the separate future auxiliary fog:

Holnap jól fog aludni. – He/She will sleep well tomorrow.
Ma este jól fog aludni. – He/She will sleep well this evening.

So grammatically it’s present tense; future meaning is possible only from context, but it’s not the default reading here.

How would I say “He does not sleep well today” using this pattern?

You insert the negation nem before the verb phrase:

Ő ma nem alszik jól. – He/She is not sleeping well today.

Position:

Ő – subject
ma – today
nem alszik jól – does not sleep well

You could also say:

Ma nem alszik jól. – Today he/she is not sleeping well. (subject implied)

Both are perfectly natural.