Une mouette vole au-dessus de la falaise, et ma nièce la montre du doigt.

Une mouette means a seagull, not the seagull. French uses the indefinite article un / une when introducing something that is not already known or identified.

So:

• une mouette = a seagull
• la mouette = the seagull

Here, the bird is being introduced for the first time, so une is the natural choice.

Vole is the 3rd person singular present tense of voler (to fly).

The subject is une mouette, which is singular, so the verb must also be singular:

• je vole = I fly
• tu voles = you fly
• il / elle vole = he / she flies
• nous volons = we fly
• vous volez = you fly
• ils / elles volent = they fly

So Une mouette vole means A seagull is flying or A seagull flies, depending on context.

Au-dessus de means above or over.

It is a fixed expression:

• au-dessus de la falaise = above the cliff

Literally, it is built from:

• au-dessus = above
• de = of / from

In French, you normally say au-dessus de + noun:

• au-dessus de la table = above the table
• au-dessus de la mer = above the sea
• au-dessus de la falaise = above the cliff

English speakers often want to say something like above the cliff directly, but in French the de is required.

Because falaise is a feminine singular noun, and after de + la, you get de la.

So:

• le
  • de becomes du
• la stays de la
• les
  • de becomes des
• before a vowel sound, de l’

Examples:

• au-dessus du toit = above the roof
• au-dessus de la falaise = above the cliff
• au-dessus des arbres = above the trees
• au-dessus de l’eau = above the water

Because nièce is a feminine noun, so the possessive adjective must agree with it:

• mon neveu = my nephew
• ma nièce = my niece

French possessive adjectives agree with the thing possessed, not with the owner.

Here, the possessed person is nièce, which is feminine singular, so ma is correct.

Here, la is a direct object pronoun meaning it. It refers back to une mouette.

So:

• Une mouette... = A seagull...
• ma nièce la montre... = my niece points it out / shows it...

French often replaces a repeated noun with a pronoun. Instead of saying:

• ma nièce montre la mouette du doigt

French can say:

• ma nièce la montre du doigt

Important point: in French, object pronouns usually come before the conjugated verb:

• je le vois = I see him / it
• elle la montre = she points to it / shows it
• nous les regardons = we watch them

Yes, la montre can mean the watch in other contexts, because une montre is a watch as a noun.

But in this sentence, la montre is not a noun phrase. It is:

• la = object pronoun (it)
• montre = verb form of montrer (shows / points out)

You can tell from the structure:

• ma nièce la montre du doigt

This clearly has a subject (ma nièce) followed by a pronoun (la) and a verb (montre).

So here it means something like my niece points at it, not my niece the watch...

Because mouette is feminine, so the direct object pronoun must also be feminine singular:

• le = him / it (masculine singular direct object)
• la = her / it (feminine singular direct object)
• les = them (plural direct object)

Since une mouette is feminine, it becomes la.

And lui is different: it is usually an indirect object pronoun meaning to him / to her. It would not be used here.

Montrer du doigt is an idiomatic expression meaning to point at or to point out with your finger.

So:

• ma nièce la montre du doigt = my niece points at it

Literally, it is something like shows it with the finger, but in natural English the best translation is usually points at it.

This is a common French expression.

Because montrer du doigt is a fixed idiomatic expression.

The du here is the contraction of de + le:

• de le doigt → du doigt

Even though it may look strange from an English point of view, this is just how the expression works in French. You should learn it as a chunk:

• montrer quelqu’un du doigt = to point at someone
• montrer quelque chose du doigt = to point at something

You may also hear pointer du doigt, which is very similar.

Not always. In French, a comma before et is often omitted, just like in English, especially in simple sentences.

So both of these can be acceptable depending on style:

• Une mouette vole au-dessus de la falaise et ma nièce la montre du doigt.
• Une mouette vole au-dessus de la falaise, et ma nièce la montre du doigt.

The comma can help separate the two clauses and make the sentence easier to read, but it is not strictly required in every case.

Yes. The French present tense can often translate in more than one way in English, depending on context.

So vole and montre can be understood as:

• flies / points
• is flying / is pointing

That means the sentence could be translated naturally as either:

• A seagull flies above the cliff, and my niece points at it.
• A seagull is flying above the cliff, and my niece is pointing at it.

French does not always make the same present-time distinction that English does between simple present and present progressive.