La donna dice che gennaio è il mese più lungo dell’inverno.

La is the definite article, meaning the.

So la donna = the woman.

In Italian, articles are used very often, sometimes more often than in English. Here, the sentence is talking about a specific woman, so la donna is the natural choice.

• donna = woman
• la donna = the woman

In Italian, the names of months are normally written with a lowercase letter, unlike in English.

So:

• gennaio = January
• febbraio = February
• marzo = March

This is standard in Italian unless the word happens to begin a sentence.

Che here means that and introduces a new clause:

• La donna dice = The woman says
• che gennaio è il mese più lungo dell’inverno = that January is the longest month of winter

So dice che... is a very common structure in Italian, just like says that... in English.

Italian usually keeps che in this kind of sentence, even where English might sometimes leave out that:

• English: She says (that) January is...
• Italian: Dice che gennaio è...

Because è is the verb is from essere.

• è = is
• e = and

The accent is important because it distinguishes two different words:

• gennaio è freddo = January is cold
• gennaio e febbraio = January and February

So in your sentence, è means is.

Month names in Italian usually appear without an article when you are simply naming the month.

So:

• gennaio = January
• in gennaio = in January
• gennaio è freddo = January is cold

You may sometimes see an article with months in special contexts, but in a basic sentence like this, no article is needed.

This is the Italian way to say the longest month.

It uses the pattern:

definite article + noun + più + adjective

So:

• il mese = the month
• più lungo = longer / longest, depending on context

Together:

• il mese più lungo = the longest month

This is called a relative superlative. Italian often expresses the longest as the month more long, literally.

Because lungo has to agree with mese, which is masculine singular.

• il mese is masculine singular
• so the adjective is lungo

Compare:

• il mese lungo = the long month
• la giornata lunga = the long day
• i mesi lunghi = the long months
• le giornate lunghe = the long days

In your sentence, lungo matches mese.

By itself, più lungo can mean longer.

But in this sentence it appears in the structure:

il mese più lungo dell’inverno

That means the longest month of winter.

The definite article il plus the noun mese makes it clear that we are choosing one month as the one with the greatest length within a group.

Compare:

• gennaio è più lungo di febbraio = January is longer than February
• gennaio è il mese più lungo dell’inverno = January is the longest month of winter

So:

• più + adjective + di often gives a direct comparison: longer than
• il/la + noun + più + adjective + di / del / della... often gives a superlative: the longest / the biggest / the most...

Because di + il becomes del, and before a vowel this becomes dell’.

Step by step:

• di + il inverno would normally contract to del inverno
• but inverno begins with a vowel, so Italian uses elision:
• del inverno becomes dell’inverno

So:

• dell’inverno = of the winter / of winter

This is a very common pattern:

• dell’estate = of the summer
• dell’anno = of the year
• dell’amico = of the friend

Yes, literally it is of the winter, but in natural English we usually say of winter or simply make it sound smoother as part of the whole phrase.

So:

• il mese più lungo dell’inverno
  literally: the month most long of the winter
  natural English: the longest month of winter

Literal meanings can help you see the grammar, but the natural translation is usually less word-for-word.

Normally, no. In standard Italian, che is expected after dire when it introduces a full clause.

So the normal form is:

• La donna dice che gennaio è il mese più lungo dell’inverno.

Without che, it would sound wrong or at least very unnatural in standard usage.

English is more flexible here:

• She says that January is...
• She says January is...

Italian normally keeps che.

The structure is:

La donna | dice | che gennaio è il mese più lungo dell’inverno

Breaking it down:

• La donna = subject
• dice = verb
• che... = subordinate clause
• gennaio = subject of the subordinate clause
• è = verb
• il mese più lungo dell’inverno = complement

So the sentence is built very similarly to English:

• The woman says
• that January is the longest month of winter

That makes this a nice example of a fairly straightforward Italian sentence with a few important grammar points inside it.