Learn German: Du arbeitest sehr gut.

The base verb is arbeiten (to work) and its stem is arbeit-. If you just added the regular informal 'you' ending -st, you would get arbeitst, which is very hard to pronounce! To make it easier to say, German adds an extra e before the ending, making it arbeitest.

How would I say this if I were speaking to my boss or a stranger?

You would use the formal 'you', which is Sie, instead of the informal du. The formal verb ending is -en, so the sentence would become: Sie arbeiten sehr gut.

The translation says 'well', but doesn't gut mean 'good'?

Yes, gut means 'good'. In English, we often add '-ly' to change a describing word into an action-describing word, or we change it entirely (like 'good' to 'well'). In German, you don't have to do that! You can just use the base word gut to mean both 'good' and 'well'.

Could I change the word order to match English better, like Du sehr gut arbeitest?

No, because German has a strict rule for statements: the conjugated verb must always be in the second position. The action word (arbeitest) has to stay right after the subject (Du), pushing sehr gut to the end.