Learn Finnish: Tämä tuoli on uusi, mutta se on halpa. - This chair is new, but it is cheap.

In Finnish, you almost always put a comma before mutta (but) when it connects two complete thoughts. Because both Tämä tuoli on uusi (This chair is new) and se on halpa (it is cheap) have their own subject and verb, they are complete sentences joined by mutta, so a comma is required.

Do I have to say se on in the second part, or can I just say mutta halpa?

You can absolutely just say mutta halpa (but cheap). If you do that, you drop the comma because the second part isn't a full sentence anymore: Tämä tuoli on uusi mutta halpa. The example sentence includes se on to show how mutta connects two full sentences.

Why does the second part use se instead of hän?

In standard Finnish, hän means he or she and is strictly used for humans. Because a chair (tuoli) is an object, we must use the pronoun se, which means it.

Why are uusi and halpa in their basic forms, instead of the partitive case?

In an 'A is B' sentence, the adjective describes the subject. Because a chair (tuoli) is a single, countable object, the adjectives describing it stay in their basic dictionary form. You would only use the partitive case here if you were describing something uncountable, like water or coffee.