Learn Russian: А клиент видит директора?

In this sentence, the director is the direct object, meaning he is the person being seen. Since директор is a masculine, living (animate) noun, its Accusative case ending changes by adding -а. For masculine animate nouns, the Accusative form looks exactly like the Genitive case.

Why didn't клиент change its ending if it's also a masculine, living person?

Клиент is the subject of the sentence (the person doing the seeing), so it stays in the Nominative case, which is its basic dictionary form. We only use the Accusative case for the object receiving the action.

Why does the sentence start with А for "And" instead of И?

The word А is used here to shift focus or ask a follow-up question, similar to "And what about...?" or "But..." in English. The word И is usually just used to connect things together, like items in a simple list.

The dictionary form of the verb "to see" is видеть. Why does it conjugate to видит with an -и-?

The verb видеть is an exception. Even though its infinitive ends in -еть, it follows the second conjugation pattern (the "-и- type" verbs). That is why the "he/she/it" form is видит, not видет.