well
The German word 'gut' is used in the same way as the English word 'well'. It is used to express that something is done to a high standard or with a high level of skill. For example, 'Sie spielt Klavier gut' means 'She plays the piano well'.
He speaks German well.
This phrase is used to appreciate someone's language skills. Here, 'gut' means 'well' and is used as an adverb modifying the verb 'spricht' (speaks).
The weather is good.
In this case, 'gut' is used to describe the weather. In English, one might translate 'gut' as 'fine' or 'nice', depending on the specifics of the weather situation.
I feel good.
This sentence is when you want to express that you are doing well, mainly with regard to your health or mood.