Paul souhaite un joyeux Noël à Marie.

Souhaite is the third-person singular present indicative form of the verb souhaiter.

The subject is Paul, which is a third-person singular subject, so French uses:

• je souhaite
• tu souhaites
• il / elle / on souhaite
• nous souhaitons
• vous souhaitez
• ils / elles souhaitent

So Paul souhaite means Paul wishes.

In this sentence, souhaiter follows the pattern:

souhaiter quelque chose à quelqu’un
= to wish something to someone

So the parts are:

• Paul = the person doing the wishing
• un joyeux Noël = the thing being wished
• à Marie = the person receiving the wish

That structure is very common in French.

That is a very common learner question.

When French uses souhaiter, it often treats the greeting as a noun phrase:

• souhaiter un joyeux Noël à quelqu’un
• literally: to wish a merry Christmas to someone

So un is normal here.

But if you are simply saying the greeting by itself, French usually says:

• Joyeux Noël !

without un.

So:

• Joyeux Noël ! = the greeting itself
• Je te souhaite un joyeux Noël. = I wish you a Merry Christmas

Because Marie is the person receiving the wish, and with souhaiter, that person is introduced by à.

Again, the pattern is:

souhaiter quelque chose à quelqu’un

Examples:

• Je souhaite bonne chance à mon frère.
• Nous souhaitons un bon anniversaire à Julie.

So in your sentence, à Marie means Marie is the indirect object: the person to whom the wish is directed.

Yes, that is also correct.

Both of these are grammatical:

• Paul souhaite un joyeux Noël à Marie.
• Paul souhaite à Marie un joyeux Noël.

The meaning is the same.

The version with à Marie earlier can sound a little more balanced or natural in some contexts, especially if the sentence is longer. But both are standard.

Because Marie is a proper noun, and French normally does not use an article before a person’s first name in standard French.

So you say:

• Paul
• Marie
• Sophie aime Marc

not:

• le Paul
• la Marie

In some regional or informal varieties, you may hear articles before names, but that is not the standard pattern learners should use here.

Noël is the name of the holiday Christmas, so it is capitalized.

In French, names of holidays are commonly capitalized:

• Noël
• Pâques

So un joyeux Noël keeps the capital letter because Noël is still the holiday name.

The two dots are called a diaeresis in English and a tréma in French.

In Noël, the ë shows that the vowels are pronounced separately rather than blending into one sound.

So Noël is pronounced roughly like:

no-EL

not as a single syllable like nole.

That mark helps the reader know how to pronounce the word correctly.

Joyeux is an adjective meaning joyful or merry, and in this fixed expression it comes before Noël:

• joyeux Noël

This is simply the normal established phrase for Merry Christmas.

You do not normally rearrange it.
So learners should memorize:

• Joyeux Noël !
• un joyeux Noël

as standard expressions.

Because Noël is a masculine noun, so the adjective must agree with it.

• masculine singular: joyeux
• feminine singular: joyeuse

Since it is Noël, you say:

• un joyeux Noël

If the noun were feminine, the adjective form would change.

Yes, but it is a little more like a full sentence than a simple greeting.

French speakers very often just say:

• Joyeux Noël !

If they want to express the idea in a full sentence, they may say things like:

• Paul souhaite un joyeux Noël à Marie.
• Je te souhaite un joyeux Noël.
• Nous vous souhaitons un joyeux Noël.

So your sentence is perfectly normal, especially in writing, narration, cards, messages, or more complete spoken sentences.

Yes. If Marie has already been mentioned, French would usually replace à Marie with an indirect object pronoun.

For example:

• Paul lui souhaite un joyeux Noël.

Here, lui means to her or to him, depending on context.

So:

• Paul souhaite un joyeux Noël à Marie.
• Paul lui souhaite un joyeux Noël.

Both are correct; the second is what French often prefers once the person is already known from context.

A careful approximate pronunciation is:

Paul souhaite un joyeux Noël à Marie
≈ Pol swet uh(n) zhwa-yuh No-el ah Ma-ree

A few useful points:

• Paul sounds roughly like Pol
• souhaite sounds roughly like swet
• joyeux sounds roughly like zhwa-yuh
• Noël has two clear parts: No-el
• Marie is Ma-ree

Also, in natural speech, French flows smoothly from one word to the next, so it may sound more connected than the spelling suggests.