Learn Russian: Старый кот умирал.

The imperfective aspect here shows that the action was an ongoing process in the past. It translates best to 'was dying'. If we used the perfective verb for this action (умер), it would mean the action was a completed event: 'The old cat died'.

How does the word кот affect the ending of the verb умирал?

In the past tense, Russian verbs must match the gender of the subject. Because кот (a male cat) is a masculine noun, the verb takes the masculine past tense ending -л. If the subject were feminine, the verb would end in -ла.

Why does the adjective старый end in -ый?

Adjectives have to match the gender, number, and case of the noun they describe. Since кот is a masculine, singular noun in the nominative (subject) case, the adjective takes the standard masculine ending -ый.

I usually see the word кошка for cat. Why is кот used here?

Кошка is the general word for a cat, but it also specifically means a female cat. Кот specifically means a male cat. This distinction is important because using a male cat (кот) is what makes both the adjective (старый) and the verb (умирал) take masculine endings.