Der Kaffee ist gratis.

Because Kaffee is a masculine noun in German. Its basic article is der.

• der Kaffee = the coffee
• ein Kaffee = a coffee

This is something you usually have to learn together with the noun: not just Kaffee, but der Kaffee.

In German, all nouns are capitalized. Since Kaffee is a noun, it must begin with a capital letter.

Compare:

• der Kaffee = the coffee
• gratis = free, free of charge

Only Kaffee is a noun here, so only that word is capitalized.

Ist is the third-person singular form of the verb sein, which means to be.

So:

• ich bin = I am
• du bist = you are
• er/sie/es ist = he/she/it is

In Der Kaffee ist gratis, ist means is.

This is a normal German main clause. German usually puts the conjugated verb in the second position.

So the structure is:

• Der Kaffee = subject
• ist = verb
• gratis = complement

This is very similar to English: The coffee is free.

Here, gratis is used like a predicate adjective/adverbial expression, meaning free of charge.

It comes after ist and describes the coffee:

• Der Kaffee ist gratis. = The coffee is free.

Unlike many German adjectives, gratis does not take adjective endings here, because it is not placed directly before a noun.

Because gratis is not being used before a noun. In German, adjectives usually only take endings when they come directly before a noun.

Compare:

• Der Kaffee ist gratis.
  No ending, because gratis comes after ist.

• der gratis Kaffee
  This sounds unnatural in German. People would normally say der kostenlose Kaffee instead.

So in this sentence, gratis stays unchanged.

Almost. Both can mean free of charge, but there is a small difference in tone and usage.

• gratis = free, free of charge; often a bit more informal or promotional
• kostenlos = free of charge; very common and neutral

Examples:

• Der Kaffee ist gratis.
• Der Kaffee ist kostenlos.

Both are correct, but kostenlos is often the more standard everyday choice.

No. In German, gratis only means free of charge.

If you want to say someone is free as in available, you would use something else, for example:

• Ich bin frei. = I am free.
• Bist du heute Abend frei? = Are you free this evening?

So gratis is about price, not availability.

A simple English-friendly guide would be:

dair KAH-fay ist GRAH-tis

A few notes:

• der sounds roughly like dair, but shorter and lighter
• Kaffee has stress on the second syllable: ka-FEE
• gratis is usually stressed on the first syllable: GRA-tis

Yes, depending on context.

• Kaffee can mean coffee in general:
  Kaffee ist beliebt. = Coffee is popular.
• It can also refer to a serving of coffee:
  Ein Kaffee, bitte. = A coffee, please.

In Der Kaffee ist gratis, it most naturally means the coffee being offered is free.

Yes, but the meaning changes slightly.

• Der Kaffee ist gratis. = The coffee is free.
  This often refers to a specific coffee, or coffee being offered in a particular situation.
• Ein Kaffee ist gratis. = A coffee is free.
  This can mean one coffee is free, for example as part of an offer.

So both are correct, but der and ein do different jobs.

It is neutral and works in both formal and informal contexts. Nothing in Der Kaffee ist gratis is especially casual or especially formal.

If you wanted a slightly more standard or official-sounding version, you might also see:

Der Kaffee ist kostenlos.

But gratis is perfectly normal and common.