Learn Hardcore Italian: Gli studenti fanno domande alla maestra.

fanno domande: they ask <a class="border-b border-dotted border-blue-400 dark:border-blue-500 cursor-pointer text-inherit no-underline" href="/grammar/italian/questions/overview">questions (general statement)
fanno delle domande: they ask some questions (an unspecified subset; a bit more “some”) <a class="border-b border-dotted border-blue-400 dark:border-blue-500 cursor-pointer text-inherit no-underline" href="/grammar/italian/negation/neanche-neppure-nemmeno">Neither is wrong; choose based on emphasis.

Because gli is the masculine plural definite article used before nouns starting with a vowel or certain consonant clusters: z, s+consonant, gn, ps, x, pn. Since studenti starts with s+consonant (st), you say gli studenti. Compare:

Masculine plural before most consonants: i ragazzi
Masculine plural before s+consonant, z, etc.: gli studenti, gli zaini, gli psicologi Singular follows the same pattern: lo studente (not il studente).

What is fanno and how does fare conjugate?

Fanno is the 3rd person plural (they) present tense of the irregular verb fare (to do/make). Present tense:

io faccio
tu fai
lui/lei fa
noi facciamo
voi fate
loro fanno

Why say fare domande instead of chiedere domande?

In Italian, “to ask a question” is expressed with the light-verb construction fare una domanda (also porre una domanda, more formal). You don’t say chiedere una domanda. Use:

Fare una domanda a qualcuno = ask someone a question
Chiedere qualcosa a qualcuno = ask someone for something / ask someone something