Learn Hardcore Icelandic: Ég býð þér kaffi.

Because the verb bjóða “to offer/invite” takes the recipient in the dative case: the pattern is bjóða e-m e-ð (“offer someone something”). The 2nd person singular pronoun declines like this:

nominative: þú (subject)
accusative: þig (direct object)
dative: þér (indirect object = “to/for you”)
genitive: þín In this sentence the recipient is “you,” so we use the dative: þér.

Do I need a preposition for “to you,” like “til þín” or “fyrir þig”?

No. Icelandic typically expresses “to/for someone” with the dative case alone, without a preposition, when the verb requires it. With bjóða, the recipient is dative: bjóða þér.

til þín means “to your place/toward you” (movement/goal).
fyrir þig means “for you” in the sense “on your behalf/for your benefit,” not the recipient of an offer. So here, just use þér.

Why is there no word for “a” before kaffi?

Icelandic has no indefinite article (“a/an”). Bare nouns can be indefinite by default. If you need a definite meaning, Icelandic uses a suffixed article:

kaffi = “coffee” (indefinite/mass)
kaffið = “the coffee” (definite, specific coffee already known in context)

What case is kaffi in here?

It’s the direct object, so accusative. For many neuter nouns, nominative and accusative look the same, which is why kaffi doesn’t change form. Useful forms:

nom/acc sg: kaffi
dat sg: kaffi
gen sg: kaffis
definite nom/acc sg: kaffið

Why is it býð and not bjóða or býður?

bjóða is the infinitive. The finite verb must agree with the subject. With ég (I), present tense is býð:

present: ég býð, þú/hann/hún/það býður, við bjóðum, þið bjóðið, þeir/þær/þau bjóða
past: ég bauð, við buðum
past participle/supine: boðið The vowel changes (bjóða → býð → bauð → boðið) are a regular strong-verb pattern.

How else can I naturally offer coffee?

Common, polite options:

Má bjóða þér kaffi? “May I offer you coffee?”
Má ég bjóða þér kaffi? (same, slightly fuller)
Viltu kaffi? “Would you like coffee?”
Viltu fá kaffi? “Would you like to get/have coffee?” All are idiomatic when offering someone a cup right now.

What’s the difference between bjóða, bjóða upp á, and bjóða í?

bjóða e-m e-ð: offer someone something. Example: Ég býð þér kaffi.
bjóða e-m upp á e-ð: also “offer/provide” (very common with food/drink and services). Example: Ég býð þér upp á kaffi.
bjóða e-n í e-ð: “invite someone to” an event/occasion. Example: Ég býð þér í kaffi = “I invite you for coffee (come over).” Note: bjóða upp (without “á”) can also mean “to auction.”

How would I say it to more than one person?

Use the 2nd person plural dative ykkur:

Ég býð ykkur kaffi. “I offer you (pl.) coffee.”

Is þér a formal “you”?

No. Here þér is simply the dative singular of þú. Icelandic once had a formal-address system (e.g., Þér, yður) but it’s archaic/rare today. In everyday modern Icelandic, þér in this sentence is not a special polite form; it just means “to you (sg.).”

Can I omit Ég?

Generally no. Icelandic is not a “pro-drop” language; subject pronouns are normally expressed in finite clauses. You can, however, change the construction to something like Má bjóða þér kaffi?, which doesn’t mention the subject explicitly but is idiomatic.

Can I say Ég býð kaffi þér?

Standard word order puts the indirect object before the direct object: S V IO DO → Ég býð þér kaffi. Saying Ég býð kaffi þér is unusual and would sound marked or odd unless you’re doing special emphasis/focus. Stick with þér before kaffi.

How do I negate it or ask a yes/no question?

Negation: place ekki after the verb (and before the direct object): Ég býð þér ekki kaffi.
Yes/no question: put the finite verb first: Býð ég þér kaffi?

How do I say “I invite you for coffee (to my place)”?

Use bjóða e-n í e-ð:

Ég býð þér í kaffi. = “I invite you for coffee” (come over). You can be more explicit:
Ég býð þér í kaffi heim til mín. = “I invite you over to my place for coffee.”

How do I pronounce the sentence?

Approximate guide (IPA + rough English hints):

Ég [jɛːɣ] ≈ “yeh” plus a soft, fricative “g” at the end.
býð [biːð̥ ~ biːð] ≈ “bee-th” (usually like the th in “this”; word-final ð can sound a bit devoiced).
þér [θjɛːr] ≈ “th-YAIR” (th as in “thin”; é is like “yeh”).
kaffi [kʰa.fɪ] ≈ “KAH-fi” (aspirated k; double f is long/strong). In fast speech, býð þér often simplifies so you mainly hear the þ of þér (the final ð of býð weakens before þ).

How do I say “a coffee” like in English (a cup of coffee)?

Use a cup word:

Ég býð þér einn kaffibolla. = “I offer you a cup of coffee.” (accusative: einn kaffibolla) You can also say: bolli af kaffi = “a cup of coffee” (literally “cup of coffee”).

What common mistakes should I watch out for with this sentence?

Using the wrong case for “you”: don’t say þig; use þér (dative) after bjóða.
Adding a preposition for the recipient: not til þín/fyrir þig here—just þér.
Using the wrong verb form with ég: it must be býð, not býður/bjóða.
Overusing the definite form of a mass noun: kaffi (indefinite) is normal when offering; kaffið means “the coffee” (some specific, already-identified coffee).
Confusing bjóða (offer/invite) with biðja (ask/request). For example, Ég bið þig = “I ask you,” not “I offer you.”