هذه مدرسة.

A common pronunciation is:

• هذه = hādhihi (often simplified in careful speech to something like hāði-hi)
• مدرسة = madrasa(h)

With full case endings (formal, fully vowelled reading):

• هذه مدرسةٌ = hādhihi madrasatun
  In pause (at the end of the sentence), the final -un is not pronounced, and ة is typically heard as -a(h): madrasa(h).

In Modern Standard Arabic, the present-tense verb to be is usually not stated. This is a verbless (nominal) sentence:

• هذه = the subject (مبتدأ)
• مدرسة = the predicate (خبر)

So هذه مدرسة literally works as This (is) a school.

Because مدرسة (school) is grammatically feminine in Arabic, and the demonstrative must match:

• هذا = this (masculine singular)
• هذه = this (feminine singular)

Since مدرسة is feminine, you say هذه مدرسة.

Many feminine nouns end with ة (taa marbūṭa), as in مدرسة. This ending is a strong clue that the noun is feminine (though there are exceptions in Arabic overall).

ة is called taa marbūṭa. Its pronunciation depends on context:

• When pausing (end of an utterance): usually -a(h) → madrasa(h)
• When linking to the next word in connected speech (or with grammar endings): it can sound like t
  Example in an iḍāfa (possession): مدرسةُ البنين (madrasatu l-banīn) = the boys’ school

Because the meaning is a school (indefinite), not the school (definite). In Arabic:

• مدرسة = a school
• المدرسة = the school

So هذه مدرسة is typically This is a school (introducing/identifying something as a school).

هذه المدرسة makes المدرسة definite:

• It often means This is the school (a specific, known school), or This school depending on context. Grammatically, you now have a demonstrative + definite noun phrase.

In fully vowelled MSA, yes, the predicate is typically marked as indefinite with tanwīn:

• هذه مدرسةٌ (hādhihi madrasatun)

In normal unvowelled writing, the tanwīn is usually not shown:

• هذه مدرسة (same sentence, just without short vowels/ending marks)

Also, when you stop at the end, the -un is not pronounced anyway.

• هذه: demonstrative pronoun, feminine singular; functions as the subject (مبتدأ)
• مدرسة: noun; functions as the predicate (خبر) describing/identifying the subject

This is a standard subject–predicate nominal sentence pattern in Arabic.

In MSA, the most neutral order is هذه مدرسة.
A reversed order like مدرسةٌ هذه can exist in more literary/stylistic contexts for emphasis, but it’s not the typical beginner pattern. For everyday straightforward identification, stick to هذه مدرسة.

The letter ذ (dhāl) is pronounced like the th in this (voiced), not like the th in think (voiceless).
So هذه contains that voiced th sound.

Use the feminine singular that:

• تلك مدرسةٌ (fully vowelled)
• تلك مدرسة (unvowelled)

تلك matches feminine singular nouns like مدرسة.