Quan arribi la primera convidada, la meva mare la farà passar al menjador.

Because after quan referring to a future event, Catalan normally uses the present subjunctive, not the future and not the plain present.

So:

• quan arribi = when she arrives
• arribi is the present subjunctive of arribar

This is very common in Catalan:

• Quan vingui, parlarem. = When he/she comes, we’ll talk.
• Quan acabi la classe, marxarem. = When class finishes, we’ll leave.

For an English speaker, this can feel unusual because English just uses the present: when she arrives. Catalan uses the subjunctive in this kind of future-dependent clause.

Arribi is the 1st person singular / 3rd person singular present subjunctive form of arribar.

In this sentence it means she arrives, because the subject is la primera convidada.

So the structure is:

• quan = when
• arribi = she arrives

Even though the verb form can also match I arrive in other contexts, here the subject clearly shows it means she.

Convidada means female guest or female invitee.

It comes from convidar = to invite.

Catalan often uses forms like:

• convidat = invited male person / male guest
• convidada = invited female person / female guest

In this sentence, la primera convidada means the first female guest. If the guest were male, it would be el primer convidat.

In Catalan, many common adjectives like primer / primera can come before the noun.

So:

• la primera convidada = the first guest

This order is normal and natural. In fact, with ordinal numbers such as primer, segon, tercer, putting them before the noun is very common.

That la is a direct object pronoun meaning her.

It refers back to la primera convidada.

So:

• la primera convidada = the first female guest
• la farà passar = my mother will have her come in / will show her in

Catalan often repeats the object with a pronoun even when the noun has already appeared earlier, especially when the noun is in a previous clause and is being referred to again.

Fer passar is a very common expression. Literally it is to make pass, but in natural English here it means:

• to let someone in
• to show someone in
• to usher someone in

So la farà passar al menjador means that the mother will bring the guest into the dining room / sitting room area.

This is a causative structure:

• fer + infinitive

Examples:

• La farà seure. = She will make her sit down / have her sit down.
• El va fer entrar. = He made him come in / had him come in.

Because Catalan is using a causative construction:

• fer
  • infinitive

Here:

• farà = will make / will have
• passar = come in / go through / pass

Together, farà passar means something like will have her come in or more naturally will show her in.

This is different from simply saying the guest enters by herself. The sentence focuses on the mother causing or facilitating that action.

Al is the contraction of:

• a
  • el = al

So:

• al menjador = to the dining room / into the dining room

Menjador literally means dining room, though in some contexts room names do not always match English homes perfectly. Still, dining room is the standard meaning.

Other similar contractions:

• a + la = a la
• de + el = del

Because the main action is in the future:

• la meva mare la farà passar al menjador = my mother will show her into the dining room

The sentence describes something that will happen once a future condition is met:

1. When the first guest arrives...
2. ...my mother will show her into the dining room.

So the time relationship is:

• subordinate clause: quan arribi = when she arrives
• main clause: la farà passar = she will show her in

Yes. Catalan can also say:

• La meva mare la farà passar al menjador quan arribi la primera convidada.

This means the same thing.

Starting with Quan arribi... puts the time condition first, which is very natural. The comma is normal when that clause comes first.

Yes, both are possible, depending on dialect and style.

• la meva mare = fully explicit, standard, widely understood
• ma mare = also common in many varieties of Catalan, especially in more informal or regional usage

For learners, la meva mare is a very safe and standard form to use.

Actually, English also does not normally use the future after when in this kind of sentence. English says:

• When the first guest arrives, my mother will show her in.

Not:

• When the first guest will arrive...

Catalan behaves similarly in that sense, but it goes one step further by often using the subjunctive in future-oriented subordinate clauses:

• Quan arribi...

So the similarity is that neither language uses a normal future form after when/quan here. The difference is that Catalan uses the subjunctive, while English uses the ordinary present.