Learn Russian: Он тихий и спокойный.

In Russian, the present tense of the verb "to be" (am, is, are) is usually dropped entirely. So, Он тихий literally means "He quiet", but it translates perfectly to "He is quiet".

Both adjectives describe a masculine "he", so why does тихий end in -ий while спокойный ends in -ый?

Usually, standard masculine adjectives end in -ый, like спокойный. However, Russian has a strict spelling rule: after certain letters including х, you can never write the letter ы. Because тихий has the letter х right before the ending, it must take -ий instead.

Is the first "o" in спокойный pronounced like an "o" or an "a"?

It sounds like an "a". In Russian, the letter о is only pronounced as a clear "o" when the spoken stress falls on it. In спокойный, the stress is on the second syllable, so the unstressed first о makes a short "a" sound.

How important is the й at the end of тихий and спокойный?

Very important! The letter й (short i) is a consonant that makes a "y" sound, similar to the end of the English word "boy". If you leave it off, the words sound like plurals, which changes the grammar completely.