Learn Hardcore Icelandic: Mjólkin er á tilboði í dag.

The bold -in bold is the suffixed definite article for feminine nouns, so bold mjólkin bold means bold the milk bold. Icelandic doesn’t have a separate word for the; instead, it attaches an ending:

masculine: bold -inn bold (bold maðurinn bold = the man)
feminine: bold -in bold (bold bókin bold = the book)
neuter: bold -ið bold (bold borðið bold = the table)

Why is it bold á tilboði bold and not bold á tilboð bold?

Because bold á bold takes the dative case when it indicates a location or state, and bold tilboð bold in the dative singular is bold tilboði bold. The phrase bold á tilboði bold literally means on offer and is a set way to say on sale.

Static location/state: bold á bold + dative (bold á tilboði bold = on offer)
Motion/direction: bold á bold + accusative (e.g., bold setja mjólkina á tilboð bold = put the milk on special)

What exactly is bold tilboð bold? How does it decline?

Bold tilboð bold means an offer/special deal. It’s a neuter noun. Key forms:

singular: nom/acc bold tilboð bold, dat bold tilboði bold, gen bold tilboðs bold
plural: nom/acc bold tilboð bold, dat bold tilboðum bold, gen bold tilboða bold

How is bold í dag bold put together, and why is bold dag bold in that form?

Bold í dag bold literally means in day and is the standard way to say today. With time expressions, bold í bold usually takes the accusative. The noun bold dagur bold (day) declines:

nom bold dagur bold, acc bold dag bold, dat bold degi bold, gen bold dags bold So bold dag bold here is the accusative.

Could I say bold Mjólk er á tilboði í dag bold instead of bold Mjólkin… bold? What’s the difference?

Yes.

Bold Mjólk er á tilboði í dag bold means milk (as a category) is on sale today.
Bold Mjólkin er á tilboði í dag bold usually refers to specific milk in context (e.g., at this store, the milk we’ve been discussing). Both are possible; the definite often feels more specific.

Is bold á tilboði bold the only way to say on sale? What about bold á útsölu bold, bold til sölu bold, or bold í boði bold?

They’re different:

Bold á tilboði bold: discounted/on special (item-level discount).
Bold á útsölu bold: at a sale (clearance/seasonal sale event).
Bold til sölu bold: for sale (available to buy, not necessarily discounted).
Bold í boði bold: on offer/provided/available (often complimentary or offered by a host, not a discount). For store discounts on an item, bold á tilboði bold is your go-to.

Can I change the word order, like putting bold í dag bold first?

Yes, Icelandic allows flexible word order for emphasis:

Neutral: bold Mjólkin er á tilboði í dag. bold
Time-first: bold Í dag er mjólkin á tilboði. bold
Focus on the discount: bold Mjólkin er í dag á tilboði. bold All are grammatical; the first is the most neutral.

How do I make this negative or ask it as a yes/no question?

Negative: place bold ekki bold after the finite verb bold er bold.
- bold Mjólkin er ekki á tilboði í dag. bold
Yes/no question: invert verb and subject.
- bold Er mjólkin á tilboði í dag? bold

How do I talk about yesterday or tomorrow with this sentence?

Change the time expression and, if needed, the verb tense:

Past: bold Mjólkin var á tilboði í gær. bold (The milk was on sale yesterday.)
Future: bold Mjólkin verður á tilboði á morgun. bold (The milk will be on sale tomorrow.)

Why is it bold er bold and not bold eru bold?

Bold er bold is the 3rd person singular of to be, matching the singular mass noun bold mjólkin bold (the milk). Use bold eru bold for plural subjects, e.g., bold Fernurnar eru á tilboði. bold (The cartons are on sale.)

Pronunciation tips for the whole sentence?

Approximate guidance:

bold Mjólkin bold: initial bold mj bold like m + y, bold ó bold like long “oh,” bold lk bold is a clear lk, final bold -in bold like “in.” Roughly “MYOHLK-in.”
bold er bold: “ehr.”
bold á bold: “ow” as in “cow,” but shorter.
bold tilboði bold: “TIL-bo-thi” (bold ð bold is the voiced th in “this”).
bold í bold: long “ee.”
bold dag bold: “da-gh,” with a soft, voiced g/fricative at the end. Stress is on the first syllable of each word.

Why isn’t there an English-style separate word for the?

Icelandic has no free-standing definite article. Definiteness is marked by suffixes (bold -inn/-in/-ið bold). Indefinite “a/an” doesn’t exist either; the bare noun is indefinite unless context or numerals specify otherwise.

Could I say bold á tilboðinu bold (on the offer)?

You could, but it means on the specific offer and is only natural if a particular offer has been identified (e.g., “this special bundle”). For the general retail meaning on sale, stick with bold á tilboði bold. Example of a specific reference: bold Mjólkin er á þessu tilboði. bold (The milk is on this deal/offer.)

Another natural way to express the same idea?

Yes: bold Það er tilboð á mjólk í dag. bold Literally there is an offer on milk today. This is very common in ads and signs.

If I want to refer to specific items rather than a mass noun, how would that look?

Use a countable noun and agree the verb in number:

Singular: bold Mjólkurfernan er á tilboði í dag. bold (The milk carton is on sale today.)
Plural: bold Mjólkurfernurnar eru á tilboði í dag. bold (The milk cartons are on sale today.)