Me da ternura cuando mi hija abraza su peluche y se queda dormida.

That is exactly how Spanish often expresses feelings caused by something:

• me da miedo = it scares me / it gives me fear
• me da vergüenza = it embarrasses me
• me da pena = it makes me feel sad / sorry
• me da ternura = it makes me feel tenderness / I find it adorable

So me is to me, and da is from dar. The whole phrase is a very natural Spanish way to say that something causes that feeling in you.

In this sentence, what gives the speaker that feeling is the whole situation: cuando mi hija abraza su peluche y se queda dormida.

Ternura is a warm, affectionate feeling of tenderness. In this kind of sentence, it often overlaps with ideas like:

• I find it sweet
• I find it adorable
• It melts my heart
• It makes me feel tender affection

It is not exactly the same as English tenderness in every context, but here that is the core idea.

Because unstressed object pronouns like me, te, le, nos normally go before a conjugated verb:

• me da
• te gusta
• le encanta

So me da ternura is the normal word order.

You could move other parts of the sentence around for emphasis, but me da stays together like that with a conjugated verb.

Because here cuando is a conjunction meaning when.

• cuando = when
• cuándo = when? / at what time? in direct or indirect questions

Examples:

• Cuando llega, cenamos. = When he arrives, we eat dinner.
• ¿Cuándo llega? = When does he arrive?

In your sentence, it is not a question, so it is cuando.

Spanish often uses the present tense for things that happen regularly or whenever a situation occurs.

Here the sense is something like:

• It makes me feel tender when my daughter hugs her stuffed toy and falls asleep.

This does not have to mean it is happening right now. It can describe a repeated or typical situation.

So the present tense here is very natural for:

• habits
• general truths
• things that happen whenever a certain situation occurs

Peluche usually means a stuffed toy, soft toy, or plush toy.

It does not have to be a teddy bear specifically. It could be:

• a teddy bear
• a stuffed rabbit
• a plush dinosaur
• any soft cuddly toy

In Spain, peluche is a very common everyday word for this.

Yes, in this sentence su peluche most naturally means her stuffed toy.

But grammatically, su can mean several things depending on context:

• his
• her
• its
• your (formal singular)
• their

Spanish uses su for all of those, so context tells you whose toy it is. Since the sentence is about mi hija, the natural reading is her stuffed toy.

If you wanted to make it extra clear, Spanish could also say:

• el peluche de mi hija

This is a very common structure.

• quedarse = to remain / to end up / to become
• dormida = asleep

So se queda dormida means:

• she falls asleep
• literally, something like she ends up asleep

This is different from just duerme (she sleeps / is sleeping).
Se queda dormida focuses on the change of state: she goes from being awake to being asleep.

Because the verb here is the pronominal form quedarse, not plain quedar.

• quedar and quedarse can have different meanings
• quedarse dormido/a is a set expression meaning to fall asleep

Examples:

• Me quedé dormido. = I fell asleep.
• El niño se quedó dormido. = The boy fell asleep.

Without se, the meaning would change and sound unnatural here.

Because dormida agrees with mi hija, which is feminine singular.

• mi hija → feminine singular
• therefore dormida

Compare:

• Mi hijo se queda dormido.
• Mi hija se queda dormida.
• Los niños se quedan dormidos.
• Las niñas se quedan dormidas.

This agreement is very important in Spanish when adjectives describe a person.

Because se queda dormida means she falls asleep, a change into the state of being asleep.

• dormida = asleep
• durmiendo = sleeping

These are related, but not the same.

Compare:

• Está durmiendo. = She is sleeping.
• Se queda dormida. = She falls asleep.

So the sentence is not describing the action in progress, but the moment/result of becoming asleep.

Because Spanish normally omits the subject when it is clear.

In:

• mi hija abraza su peluche y se queda dormida

the subject of both verbs is clearly mi hija, so Spanish does not need to repeat it.

English often repeats the subject more:

• my daughter hugs her stuffed toy and she falls asleep

Spanish usually avoids that unless there is a reason to emphasize or clarify.

Yes. Abraza is from abrazar = to hug.

It is a direct transitive verb, so you hug something/someone directly:

• abraza a su madre = she hugs her mother
• abraza su peluche = she hugs her stuffed toy

Notice something important:

• With people, Spanish often uses the personal a: abraza a su madre
• With things, there is no a: abraza su peluche

So abraza su peluche is exactly what you would expect.

Because dar + emotion noun often appears without an article in this kind of general expression:

• dar miedo
• dar pena
• dar vergüenza
• dar ternura

So me da ternura is the standard idiomatic form.

Adding an article, such as me da una ternura, would usually change the tone and sound much less neutral or much more expressive, and in this sentence it would not be the normal choice.

Yes, in natural English that can work very well.

Even though the Spanish literally uses ternura, the feeling is often expressed in English as:

• I find it adorable when...
• It’s so sweet when...
• It melts my heart when...
• I think it’s cute when...

The exact best translation depends on tone, but those are all close in spirit. The Spanish is affectionate and warm.

The grammar is general Spanish, but peluche is especially standard and common in Spain for stuffed toy / plush toy.

Speakers in many Spanish-speaking countries will understand it perfectly, but other regions may also use different everyday words, such as terms closer to muñeco de peluche or other local expressions.

So nothing in the sentence is unnatural outside Spain, but peluche fits Spain very naturally.