Learn Hardcore Icelandic: Mánudagurinn er rólegur í dag.

Mánudagurinn er rólegur í dag = “The Monday (today) is quiet.”
Mánudagur er rólegur í dag would sound more like “A Monday is quiet today”, which is odd and not how you talk about today’s Monday.

If you want to talk about Mondays in general, you would say for example:

Mánudagar eru rólegir. – Mondays are quiet.
Á mánudögum er rólegt. – On Mondays it is quiet.

What grammatical case is Mánudagurinn, and why is that case used?

Mánudagurinn is in the nominative case.

Reason: It is the subject of the sentence, and in Icelandic the subject of a normal sentence is in the nominative.

Subject: Mánudagurinn
Verb: er
Predicate adjective: rólegur

The adjective rólegur also appears in the nominative, because predicate adjectives agree in case (and gender, number) with the subject they describe.

Why does the adjective rólegur end in -ur here?

The ending -ur shows that rólegur is:

masculine
singular
nominative
in its strong form

It has to agree with Mánudagurinn, which is:

masculine
singular
nominative

So you get:

Mánudagurinn er rólegur. – The Monday is quiet.

Compare how rólegur changes with different genders in the same position:

Masculine: Mánudagurinn er rólegur.
Feminine: Helgin er róleg. – The weekend is quiet.
Neuter: Kvöldið er rólegt. – The evening is quiet.

Here you can see:

masculine nominative singular: rólegur
feminine nominative singular: róleg
neuter nominative singular: rólegt

How would the sentence change with a feminine or neuter subject instead of Mánudagurinn?

You change the adjective ending to match the new noun.

Feminine example:

Helgin er róleg í dag.
- helgin = the weekend (feminine)
- róleg = feminine nominative singular form

Neuter example:

Kvöldið er rólegt í dag.
- kvöldið = the evening (neuter)
- rólegt = neuter nominative singular form

Structure stays the same:

Subject (in nominative) + er + adjective (agreeing in gender/number/case) + í dag

Why do we say í dag for “today”? What does í dag literally mean?

Literally:

í = in / on
dag = day (accusative singular of dagur “day”)

So í dag is literally something like “in (this) day” or “on day”, but as a fixed expression it simply means “today”.

Important points:

í dag is one of the standard ways to say today.
There is no article (the) here – it’s not í daginn.
The noun dagur appears as dag because í here takes the accusative in time expressions.

Similar time phrases:

í gær – yesterday
á morgun – tomorrow
í kvöld – this evening / tonight

Can í dag go at the beginning or in the middle of the sentence, or must it be at the end?

It does not have to be at the end. All of these are grammatical, with slightly different emphasis:

Mánudagurinn er rólegur í dag.
– Neutral: “The Monday is quiet today.”
Í dag er mánudagurinn rólegur.
– Emphasis on today: “Today, the Monday is quiet.”
Mánudagurinn er í dag rólegur.
– Possible, but sounds more marked/emphatic; you are really stressing í dag.

Note the verb-second (V2) rule:

If you move Í dag to the front, the verb must still be in second position:
Í dag er mánudagurinn rólegur, not Í dag mánudagurinn er rólegur.

Why do we need the verb er? Could we just say Mánudagurinn rólegur í dag?

You must use er here.

er is the present tense of að vera – to be. In Icelandic, in normal sentences, you cannot drop er the way English sometimes drops is in headlines or notes.

Mánudagurinn er rólegur í dag. – Correct.
Mánudagurinn rólegur í dag. – Feels like broken / telegraphic Icelandic.

So: whenever you link a subject to an adjective or noun (“X is Y”), include the correct form of að vera.

Can I use rólega instead of rólegur? What is the difference?

No. In this sentence you cannot use rólega.

rólegur is an adjective: quiet, calm (describes a noun)
rólega is an adverb: quietly, calmly, gently (describes a verb or whole action)

You say:

Mánudagurinn er rólegur í dag. – The Monday is quiet today.
(rólegur describes Mánudagurinn)

But with an adverb you would say things like:

Hann keyrir rólega. – He drives calmly / slowly.
(rólega describes keyrir – drives)

So:

Mánudagurinn er rólegur í dag. – correct
Mánudagurinn er rólega í dag. – incorrect

Are days of the week capitalized in Icelandic?

No special capitalization is used for days of the week in Icelandic.

At the start of a sentence, the first letter is capitalized:
Mánudagurinn er rólegur í dag.
Inside a sentence, you normally write them with a lowercase first letter:
Ég kem á mánudaginn. – I’m coming on Monday.

So Mánudagurinn is capitalized here only because it is the first word of the sentence, not because it is a day name.

How is Mánudagurinn er rólegur í dag pronounced?

Very roughly, in English-like terms:

Mánu- – Má with á like the ow in now (but a bit tenser), nu like a short “nuh” with rounded lips.
-dagurinn – da like da in dark (but shorter), g is a soft g (more like the g in Spanish amigo than a hard English g), urinn like “oor-in” with a tapped r.
er – like “air” but shorter and crisper.
ró- – ró with ó similar to the vowel in go or oh, but longer.
-legur – le like le in let, gur with another soft g and a tapped r at the end.
í – like ee in see, but tenser and longer.
dag – short da plus that same soft g at the end.

Stress:

Icelandic usually stresses the first syllable of each word, so:
MÁ-nu-dagurinn er RÓ-legur í DAG (primary stress on Má- and Ró-, and on dag).