Learn Russian: Вчера директор много работал и был сердит, а сегодня он спокоен. - Yesterday the director worked a lot and was angry, but today he is calm.

These are short-form adjectives. Russian often uses short forms to describe a temporary state or mood (how he felt yesterday, how he feels today) rather than a permanent personality trait. If you used the long forms, it would sound like he is generally an angry or calm person by nature.

What would these words be if the director was a woman?

Short-form adjectives must match the gender of the person you are talking about. If the director were female, you would add an -а to the end of the adjectives: была сердита (she was angry) and она спокойна (she is calm).

Why is there an е in спокоен if the regular adjective is спокойный?

To make the masculine short form, you usually drop the regular ending (like -ый). But doing that to спокойный leaves an awkward group of consonants at the end. To make it easier to pronounce, Russian inserts a fill vowel, giving us спокоен.

Why does the sentence use был before сердит, but no verb before спокоен?

In Russian, the present tense of the verb 'to be' (am/is/are) is left out. So, 'he is calm' is simply он спокоен. However, in the past tense, you must include the verb, which is why we need был (was) before сердит.