Learn Hardcore Icelandic: Mánudagur er erfiður dagur.

The sentence Mánudagur er erfiður dagur literally means Monday is a difficult day. The second dagur is a predicate noun (a noun that renames or classifies the subject).

Mánudagur er erfiður. = Monday is difficult. (Just describing Monday.)
Mánudagur er erfiður dagur. = Monday is a difficult day. (Classifying Monday as a type of day.)

Both are grammatically correct; the version with dagur is just a bit more explicit and natural if you want to emphasize that you are talking about Monday as a day (of the week).

What case are Mánudagur, erfiður, and dagur in? Why?

All three are in the nominative singular masculine:

Mánudagur – nominative singular (subject of the sentence)
erfiður – nominative singular masculine (adjective agreeing with dagur)
dagur – nominative singular (predicate noun after er)

In Icelandic, with the verb vera (to be), both the subject and the complement (the noun/adjective after er) are normally in the nominative:

Mánudagur (nom.) er erfiður dagur (nom.).
Monday (nom.) is a difficult day (nom.).

Why does erfiður have this particular ending -ur? What is it agreeing with?

Erfiður is an adjective meaning difficult. Adjectives in Icelandic change form to agree with the noun in:

gender (masculine / feminine / neuter)
number (singular / plural)
case (nominative / accusative / dative / genitive)

The base forms for this adjective in the nominative singular (indefinite) are:

masculine: erfiður
feminine: erfið
neuter: erfitt

Since dagur is masculine singular nominative, the adjective must also be masculine singular nominative:

erfiður dagur – a difficult day
(for comparison) erfið kona – a difficult woman (feminine)
erfitt verkefni – a difficult task (neuter)

Both Mánudagur and dagur end in -ur. Are they the same kind of form?

Yes. Both Mánudagur and dagur are masculine nouns in the nominative singular, and in this class of nouns the nominative singular often ends in -ur.

dagur – day
- nominative singular: dagur
- accusative singular: dag
- dative singular: degi
- genitive singular: dags

Mánudagur is a compound based on dagur, so it has the same -ur ending in the nominative singular.

What exactly is er here, and why not eru or something else?

Er is the 3rd person singular present form of the verb vera (to be):

ég er – I am
þú ert – you are (singular)
hann / hún / það er – he / she / it is
við erum – we are
þið eruð – you are (plural)
þeir / þær / þau eru – they are

The subject Mánudagur is singular, so you use er:

Mánudagur er erfiður dagur. – Monday is a difficult day.

If the subject were plural, you would use eru:

Mánudagar eru erfiðir dagar. – Mondays are difficult days.

Er and erfiður both start with er. Is there some connection, like er erfiður = is is-difficult?

No, that’s just a coincidence in spelling:

er is the verb is (from vera).
erfiður is one word meaning difficult.

So the structure is:

Mánudagur – subject (Monday)
er – verb (is)
erfiður dagur – complement (a difficult day)

You should think of erfiður as a single unit, not as er + fiður or something similar.

Is Mánudagur always capitalized? Are names of days capitalized in Icelandic?

In Icelandic, only the first word of a sentence and proper names are capitalized. Names of days of the week are not considered proper names, so normally they are not capitalized inside a sentence:

á mánudögum – on Mondays
mánudagur er erfiður dagur. – Monday is a difficult day. (Here it’s also sentence-initial.)

In your example, Mánudagur is capitalized because it is the first word of the sentence, not because it is a weekday name.

How do you pronounce Mánudagur er erfiður dagur?

Approximate pronunciation (not strict IPA, just a learner-friendly guide):

Mánudagur – roughly MAU-nu-da-gur
- Má like mow in mow the lawn (but shorter)
- nu like ny in funny, but with a rounded u sound
- dagur like DAH-gur, with a soft g (almost like a throaty gh)
er – like air but shorter, more like ehr
erfiður – roughly EHR-vi-thur
- er = ehr
- fi = vih
- ður = th-ur, where ð is like the th in this, and ur is a short, rounded u with a light r at the end.
dagur – again, DAH-gur, as above.

Spoken together, Icelanders will reduce and blend sounds a bit, but this approximation is good enough to be understood.

Why does the adjective erfiður come before dagur? Could it go after, like in some other languages?

In Icelandic, adjectives almost always come before the nouns they modify, just like in English:

erfiður dagur – a difficult day
góður vinur – a good friend
gamalt hús – an old house

Putting the adjective after the noun is unusual and usually only happens in special stylistic or poetic contexts. For normal speech and writing, always put the adjective before the noun.

How would I say Mondays are difficult instead of Monday is a difficult day?

You have a couple of natural options:

Use the plural of Monday:
- Mánudagar eru erfiðir.
  Mondays are difficult.
Use a time expression with á (on) and the plural dative:
- Á mánudögum er erfitt.
  On Mondays it is difficult.

In your original structure, you could also pluralize everything:

Mánudagar eru erfiðir dagar. – Mondays are difficult days.