I remember the teacher said that the word "the" was added before "the weather", so why not add the word "it's a beautiful day today"? Am I wrong?

Weather, weather, is really an uncountable noun.

As a definite article, the is usually used in specific occasions. In addition, please remember that any noun in English (except proper nouns such as names and places) cannot be singular, that is, the, a (only countable nouns can be added after a), that, this or plural should be added before it.

Weather refers to the weather in a certain place.

It's a beautiful day today, because this is an exclamatory sentence that modifies the uncountable noun what weather, so there is no need to add the.