Aquest vestit té una cremallera a l'esquena i dos botons petits.

Because aquest has to agree with the noun it modifies.

• aquest = this for a masculine singular noun
• aquesta = this for a feminine singular noun

The noun vestit is grammatically masculine singular, so the correct form is aquest vestit.

Yes. In Catalan, grammatical gender does not always match any real-world idea of sex or gender.

So even if the English meaning is dress, the Catalan noun vestit is still masculine. That is why you get:

• aquest vestit
• not aquesta vestit

This is very common in Romance languages: you simply have to learn each noun’s grammatical gender.

Té is the 3rd person singular present of the verb tenir (to have):
ell/ella té = he/she/it has

The accent is part of the standard spelling. It also helps distinguish it from te, which can mean tea.

So here:

• Aquest vestit té... = This dress has...

Because this is a very natural way in Catalan to describe the features or parts of an object.

So:

• Aquest vestit té una cremallera...
  literally: This dress has a zipper...

English does the same thing very often, so this structure is straightforward:

• The coat has two pockets
• The shirt has long sleeves
• Aquest vestit té...

Because cremallera is a feminine singular noun, so it takes the feminine singular indefinite article una.

• un = a/an for masculine singular nouns
• una = a/an for feminine singular nouns

So:

• una cremallera = a zipper

The gender is grammatical, so you learn cremallera as a feminine noun.

Because the article la becomes l' before a vowel sound.

So:

• la esquena → l'esquena

And with the preposition a:

• a + l'esquena = a l'esquena

This apostrophe is very common in Catalan with articles before words beginning with a vowel.

Here it means something like at the back or on the back part of the dress.

With clothing, esquena literally means back, but in a sentence like this it refers to the back area of the garment, not necessarily a person’s physical back.

So:

• una cremallera a l'esquena
  = a zipper at the back / in the back

This is a normal, idiomatic way to describe where something is on a piece of clothing.

Because botons is masculine plural.

In Catalan, the number two has two forms:

• dos for masculine nouns
• dues for feminine nouns

So:

• dos botons = two buttons
• dues butxaques = two pockets

Since botó is masculine, the plural phrase is dos botons.

Because after a number, Catalan normally uses the noun directly, without an article.

So:

• dos botons petits = two small buttons

That is the normal pattern:

• tres llibres
• quatre cadires
• dos botons

You do not need an extra word equivalent to a/an here.

Because in Catalan, adjectives usually come after the noun.

So the normal order is:

• botons petits = small buttons

not usually:

• petits botons

Also, petits agrees with botons:

• botons = masculine plural
• petits = masculine plural

That final -s shows plural, and the masculine form is petits.

Several words are agreeing with the nouns they describe:

• aquest vestit
  aquest agrees with vestit → masculine singular

• una cremallera
  una agrees with cremallera → feminine singular

• dos botons petits
  dos and petits agree with botons → masculine plural

This is one of the most important patterns in Catalan: determiners and adjectives usually match the noun in gender and number.

A rough English-friendly pronunciation is:

uh-KEST buhs-TEET tay OO-nuh kruh-muh-YEH-ruh uh luhs-KEH-nuh ee doz boo-TONS puh-TEETS

A few notes:

• Aquest: the stress is on the second syllable
• vestit: stress on the last syllable
• té: sounds roughly like teh with a clear stressed vowel
• cremallera: the ll is a Catalan sound often like ly or y, depending on accent
• esquena: stress on que
• petits: stress on the second syllable

Pronunciation varies by dialect, but that approximation will help you say it understandably.