Mon beau-père dit que ma belle-mère a tout de suite aimé ma fiancée.

They can mean either one.

• beau-père = father-in-law or stepfather
• belle-mère = mother-in-law or stepmother

French uses the same words for both ideas, so the exact meaning depends on the context.

Also, in these family terms, beau and belle do not literally mean handsome and beautiful. They are just part of the fixed expressions.

Because French possessive adjectives agree with the gender of the noun, not with the speaker.

• beau-père is masculine, so you use mon
• belle-mère is feminine, so you use ma

So even if the speaker is female, she would still say:

• mon beau-père
• ma belle-mère

Because fiancée is a feminine noun here.

• un fiancé = a male fiancé
• une fiancée = a female fiancée

Since the noun is feminine, the possessive is ma:

• ma fiancée

The extra -e at the end shows the feminine form.

The difference is gender:

• fiancé = an engaged man / male fiancé
• fiancée = an engaged woman / female fiancée

In this sentence, ma fiancée tells you the speaker is engaged to a woman.

The accents are normal spelling here:

• fiancé
• fiancée

Because a aimé is the passé composé, which is a common French past tense.

• aime = likes / loves or is liking/loving in the present
• a aimé = liked / loved in the past

So:

• ma belle-mère aime ma fiancée = my mother-in-law/stepmother likes my fiancée
• ma belle-mère a aimé ma fiancée = my mother-in-law/stepmother liked my fiancée

Here, the idea is that she liked her right away, so the past tense makes sense.

a is the present-tense form of avoir used as an auxiliary verb.

So a aimé is built like this:

• a = has
• aimé = past participle of aimer

Literally, it is similar to has liked, but in natural English we often just translate it as liked.

Because with avoir, the past participle usually does not agree with the subject.

So even though ma belle-mère is feminine, you still say:

• elle a aimé

not something different.

In this sentence, aimé stays the same.

tout de suite means immediately, right away, or straight away.

It is a very common fixed expression. You should learn it as one chunk, because the literal meanings of the individual words do not help much here.

So:

• a tout de suite aimé = liked immediately / liked right away

Because in French, short adverbs and adverbial expressions often go between the auxiliary and the past participle in the passé composé.

So this is very natural:

• a tout de suite aimé

That word order is standard and idiomatic.

You may also see adverbs in other positions in French, but here this placement sounds very normal.

que introduces a subordinate clause. Here it means that.

• Mon beau-père dit que... = My father-in-law/stepfather says that...

In English, that is often optional:

• My father-in-law says my mother-in-law liked my fiancée right away

But in French, que is normally included after a verb like dire when introducing a full clause.

Because dit is the form for il / elle / on in the present tense of dire.

Here the subject is Mon beau-père, which is third person singular, so you use:

• mon beau-père dit = my father-in-law/stepfather says

Compare:

• je dis = I say
• tu dis = you say
• il/elle/on dit = he/she/one says

Because these are fixed compound nouns in French.

Family relationship terms like these are normally written with hyphens:

• beau-père
• belle-mère

The hyphen shows that the two parts function together as one expression.

Yes, that is understandable, but a tout de suite aimé is more natural in this sentence.

French often prefers to place tout de suite before the past participle when it modifies the whole action:

• a tout de suite aimé

Putting tout de suite later is possible in some contexts, but the original version sounds smoother and more standard here.