Feliz cumpleaños, amiga.

Because this is a fixed greeting, just like English Happy birthday! Spanish often leaves out words that are understood from context.

So Feliz cumpleaños works as a complete expression even without a verb like I wish you or have.

Cumpleaños is a special noun. It comes from the idea of cumplir años—literally, to complete/turn years. Even when talking about one birthday, the word stays cumpleaños.

So:

• el cumpleaños = the birthday
• los cumpleaños = the birthdays

The form itself does not change between singular and plural; the article or context tells you which one it is.

Because in this sentence cumpleaños is being treated as a singular noun: one birthday.

So the adjective is singular too:

• feliz cumpleaños = singular idea

If you really needed a plural adjective, feliz becomes felices, but in the standard birthday greeting to one person, feliz is correct.

Here amiga is being used for direct address—you are talking to the person directly.

In English, that is like:

• Happy birthday, friend
• Happy birthday, Maria

This use is sometimes called the vocative. It is not the object of the sentence; it just identifies who you are speaking to.

Because amiga is a direct address word. In Spanish, as in English, you normally separate direct address with commas in writing.

So:

• Feliz cumpleaños, amiga.

Without the comma, the sentence looks less natural and less correct in standard writing.

No. You can simply say:

• Feliz cumpleaños.

Adding amiga makes it more personal and affectionate. Use it only if it fits your relationship with the person.

For example:

• Feliz cumpleaños. = neutral, standard
• Feliz cumpleaños, amiga. = warm, personal, said to a female friend

Because amiga is the feminine form, used when speaking to a female friend.

• amiga = female friend
• amigo = male friend

So you would say:

• Feliz cumpleaños, amiga. to a woman
• Feliz cumpleaños, amigo. to a man

No. Feliz is the same for masculine and feminine singular nouns.

So:

• feliz cumpleaños, amigo
• feliz cumpleaños, amiga

Both use feliz.
Only the noun for the person changes: amigo/amiga.

A good approximation is koom-pleh-AHN-yos.

A few helpful points:

• the stress falls on a: cumpleAños
• ñ sounds like ny in canyon
• the vowels are clear and short, not reduced like in English

So años sounds roughly like AHN-yos, not AN-os.

Yes. Feliz cumpleaños is the standard and widely understood birthday greeting across Latin America.

You may also hear more casual or alternative expressions such as:

• Feliz cumple = a shorter, more informal version
• Muchas felicidades = many congratulations / best wishes

But if you want the safest and most standard choice, use Feliz cumpleaños.