Learn Russian: Что пишет ваш учитель?

The verb писать (to write) is an irregular verb. In the present tense, the letter с changes to ш. For example, "he/she writes" becomes пишет, and "I write" becomes я пишу.

Does пишет mean "writes" or "is writing"?

It means both! Russian doesn't have a separate verb tense for actions happening right now. The present tense verb пишет covers both "writes" in general and "is writing" at this exact moment.

I thought ч makes a "ch" sound. Why does Что sound like "shto"?

You are right about the letter ч, but the word что (what) is a very common exception. It is almost always pronounced as if it were spelled што.

The word учитель ends in a soft sign. How do we know to use the masculine ваш instead of ваша?

Nouns ending in a soft sign (ь) can be masculine or feminine, which you often just have to memorize. However, учитель specifically means a male teacher, so it is a masculine noun and takes the masculine possessive ваш. A female teacher would be учительница.