Misafirler gelince aile fotoğrafını göstereceğim.

Gelince comes from the verb stem gel- (to come / arrive) plus the suffix -ince.

This suffix makes a time clause and usually means:

• when
• once
• as soon as

So misafirler gelince means when the guests arrive.

A useful point: -ince itself does not show person. The subject misafirler tells you who is coming.

Turkish often leaves out subject pronouns when the verb ending already makes the subject clear.

Here, göstereceğim means I will show. The ending tells you the subject is I, so ben is unnecessary.

You could say Ben göstereceğim for emphasis, but in a normal sentence it is usually omitted.

-ler / -lar is the regular plural suffix in Turkish.

• misafir = guest
• misafirler = guests

The form is -ler here because of vowel harmony.

Also, Turkish does not use the the way English does, so misafirler can mean the guests or just guests, depending on context.

This is a very common learner question, because there are really two layers here.

1. aile fotoğrafı

This means family photo.

It is a Turkish noun compound:

• aile = family
• fotoğrafı = photo with the compound ending

In Turkish, noun compounds often add a 3rd person possessive-looking ending to the second noun:

• aile fotoğrafı = family photo
• literally something like family its-photo, but you should learn it as the normal Turkish way to say family photo

2. aile fotoğrafını

Then the whole thing becomes the direct object of the verb show, and it is a specific / definite object, so it takes the accusative ending.

Because the noun already ends in that compound/possessive form, the accusative appears with a buffer n:

• fotoğrafı
  • -nı → fotoğrafını

So aile fotoğrafını means the family photo as a specific object.

Usually, aile fotoğrafı is best understood as family photo.

It is a standard noun compound, not necessarily a strong possessive like the photo of the family or the family’s photo in English.

In real context, it often refers to a photo showing the family, or a family-related photograph. English and Turkish do not divide these meanings in exactly the same way, so the best translation depends on context.

Because it is a definite direct object.

In Turkish, a direct object usually takes the accusative ending when it is specific or definite.

Compare:

• Bir fotoğraf göstereceğim = I will show a photo
  indefinite, so no accusative

• Fotoğrafı göstereceğim = I will show the photo
  definite, so accusative appears

In your sentence, the speaker has a specific photo in mind, so aile fotoğrafını takes the accusative.

Göstereceğim breaks down like this:

• göster- = show
• -ecek / -acak = future tense
• -im = I

So the basic structure is:

göster + ecek + im = I will show

Because of normal sound changes in Turkish, -ecek + im becomes -eceğim in spelling and pronunciation:

• gösterecekim is not correct
• göstereceğim is correct

The k softens to ğ before the vowel-initial personal ending.

Because that is a very natural Turkish word order.

Turkish often puts:

• time expressions
• conditions
• background information

before the main action, and the main verb usually comes at the end.

So:

• Misafirler gelince aile fotoğrafını göstereceğim

feels very natural: first the situation, then what the speaker will do.

Other word orders are possible for emphasis, but this one is very standard.

Yes. The suffix -ince often covers several close English ideas:

• when
• once
• as soon as

The exact shade depends on context.

So this sentence could feel like:

• When the guests arrive, I’ll show the family photo
• Once the guests arrive, I’ll show the family photo
• As soon as the guests arrive, I’ll show the family photo

English may choose different words, but Turkish uses the same structure here.

Because gelince is a subordinate time form, not a full tense like English will come.

In Turkish, the timing relationship is often handled this way:

• the subordinate clause uses a special form like gelince
• the main clause carries the main tense, here göstereceğim = I will show

So Turkish does not need something like misafirler gelecek olunca here.
Misafirler gelince already clearly means when the guests arrive in relation to the future main action.

It depends on context.

You could say just:

• Gelince aile fotoğrafını göstereceğim

but then who is arriving would have to be understood from context.

Because gelince itself does not mark person, adding misafirler makes the meaning clear: it is the guests who are arriving.

So in your sentence, misafirler is there to identify the subject of the time clause clearly.