Learn Russian: Директор часто объявляет перерыв.

The hard sign (ъ) is used to separate a prefix ending in a consonant (like об-) from a root that starts with a softening vowel (е, ё, ю, or я). Here, it prevents the б from blending into the я, keeping the pronunciation sharply separated.

What case is перерыв in this sentence, and why doesn't its ending change?

It is in the Accusative case because it is the direct object (the thing being announced). Because перерыв is a masculine inanimate noun, its Accusative form looks exactly the same as its Nominative (dictionary) form.

Why is the imperfective verb объявляет used here instead of its perfective pair?

The presence of the adverb часто (often) indicates that this is a habitual, repeated action. Repeated actions always require the imperfective aspect. If the director were making a single, one-time announcement, you would use the perfective verb объявить.

How is the noun перерыв constructed?

It is built from the prefix пере- (across, over) and the root -рыв, which is related to tearing or breaking (as seen in the verb рвать, to tear). Quite literally, it translates to a 'break' or interruption in an activity.