Learn Hardcore Spanish (Latin America): Esa es la profesora de música a quien mi hijo admira. - That is the music teacher whom my son admires.

It means whom. The a is the personal a, which Spanish uses before a human direct object. Since the verb is admirar and the admired person is the teacher, you get a quien. Compare:

Person: Mi hijo admira a la profesora.
Thing: Mi hijo admira la música. (no personal a)

Could I use que instead of a quien? What would change?

Yes. Three natural options:

… la profesora de música que mi hijo admira. (very common and neutral)
… la profesora de música a quien mi hijo admira. (a bit more formal)
… la profesora de música a la que mi hijo admira. (also common)

Do not say a que here for the personal a: … la profesora a que mi hijo admira is not standard.

Do I need a comma before a quien?

It depends on meaning.

No comma (as given): … la profesora de música a quien mi hijo admira. Restrictive/identifying: it specifies which music teacher you mean.
With comma: … la profesora de música, a quien mi hijo admira. Non-restrictive/parenthetical: you’ve already identified her, and you’re adding extra info.

Why is it quien without an accent and not quién?

Quien (no accent) is a relative pronoun: “the one who/whom.” Quién (with accent) is interrogative/exclamative: ¿A quién admira tu hijo? In your sentence it’s relative, so no accent.

Is quien only used for people?

Practically yes. Quien/quienes refer to people (or personified beings in special cases). For things, use que or el/la cual.

What’s the difference between starting with Esa es… and Ella es…?

Esa es… points to someone you can identify in context (visually or situationally): “That (woman) is…”
Ella es… refers back to a previously mentioned female person: “She is…”

Both are fine; esa is deictic (pointing), ella is a pronoun for “she.”

Does esa ever take an accent (like ésa)?

Modern standard spelling does not require an accent: write esa. The old accented form (ésa) is now only used, very rarely, to avoid ambiguity. You’re safe with esa.

Why profesora and not profesor?

Agreement with gender. La profesora is feminine; el profesor is masculine. If the teacher were male, you’d say: Ese es el profesor de música a quien mi hijo admira.

Could I say maestra instead of profesora in Latin America?

Often, yes. In much of Latin America, maestra/maestro is common for elementary school, and profesora/profesor for middle school, high school, and university. Usage varies by country. Your sentence with maestra is natural in many places: Esa es la maestra de música a quien mi hijo admira.

Why de música and not an adjective like musical?

To express the subject a teacher teaches, Spanish typically uses de + subject: profesora de música, profesor de historia. Profesora musical would mean “a musical teacher,” which is not the intended meaning. Also, don’t use the article here: say de música, not de la música.

Could I drop the article and say Esa es profesora de música?

Yes, but the meaning changes:

Esa es la profesora de música = That is the (specific) music teacher.
Ella es profesora de música = She is a music teacher (profession in general). With ser + profession, Spanish usually omits the article when speaking non-specifically.

Why is the verb admira (3rd person singular)?

Because the subject is mi hijo (my son), which is third person singular. Hence (él) admira. If it were plural: mis hijos admiran.

How would this sentence look in the plural?

One teacher, several admirers: Esa es la profesora de música a quien mis hijos admiran.
Several teachers, several admirers: Esas son las profesoras de música a quienes mis hijos admiran. (Note the plural a quienes and admiran.)

If I split it into two sentences, which object pronoun should I use for profesora?

Use the direct object pronoun la: Mi hijo la admira. In most of Latin America, using le here (leísmo) is not standard; stick with la for a feminine direct object.

Can the word order inside the relative clause change?

Yes, Spanish allows some flexibility. All of these are acceptable and mean the same:

… que mi hijo admira
… que admira mi hijo
… a quien mi hijo admira
… a quien admira mi hijo Choose the one that sounds most natural to you; all are common in Latin America.