Learn Romanian: Profesorul vă mulțumește.

It is represented by the word vă. The word vă is a short pronoun meaning 'you' or 'to you'. Romanian uses vă as the short form for both the plural 'you' (a group of people) and the formal 'you' (dumneavoastră). Even though the full long word isn't there, vă is the correct polite short form to use here.

Why does the verb say mulțumește instead of mulțumesc?

The verb must match the person doing the action. The familiar word mulțumesc means 'I thank' (or 'they thank'). Since the teacher (profesorul) is a 'he', we must use the 'he/she' form of the verb, which is mulțumește.

Why does vă come before the verb instead of after it?

In English we say 'thanks you', putting 'you' after the action. But in Romanian, short pronouns like vă almost always jump right in front of the verb. So literally, the sentence order is 'The teacher you thanks'.

I remember the word profesor, but why does it end in -ul here?

The -ul at the end of profesorul is the Romanian way of saying 'the' for masculine nouns. Instead of putting a separate word in front like in English, you attach the 'the' directly to the end of the noun.