Learn Hardcore Finnish: Tervetuloa kurssille!

Finnish often uses fixed expressions for greetings and welcomes without a finite verb. Tervetuloa kurssille! is a complete, natural exclamation, just like English Welcome to the course! does not need a full sentence such as You are welcome to the course.

Is tervetuloa a form of the verb tulla?

Not in the way a beginner might expect. It is not the normal imperative come! In modern Finnish, tervetuloa is best learned as a fixed expression meaning welcome.

It is historically connected with the idea of coming/arrival, which is why it may look related to tulla, but for practical learning it is much better to memorize tervetuloa as one whole expression.

What is the difference between tervetuloa and tervetullut?

They are used differently:

Tervetuloa! = the greeting Welcome!
tervetullut = an adjective meaning welcome

For example:

Tervetuloa kurssille! = a greeting
Olet tervetullut kurssille. = You are welcome to the course.

So you would not normally greet someone by saying tervetullut! by itself. The usual welcoming expression is tervetuloa!