Learn Russian: Эта шапка тёплая и красивая.

In Russian, the verb "to be" is almost never used in the present tense. Instead of saying "This hat is warm," you simply state the noun and its qualities: "This hat warm" (Эта шапка тёплая).

Why do the words Эта, тёплая, and красивая all have similar endings?

They are all matching the gender of the noun they describe. Шапка is a feminine noun because it ends in -а. Therefore, the word for "this" must be the feminine Эта, and the adjectives must take the feminine ending -ая.

I often see Это used for "this". Why is it Эта here instead of Это?

If you are pointing at something and saying "This is a hat," you use the fixed default word Это: Это шапка. But in this sentence, "this" is attached directly to the noun ("This hat..."). Because it acts like an adjective pointing to the feminine noun шапка, it has to change its form to match the feminine Эта.

How would the Russian change if I wanted to say "This is a warm and beautiful hat"?

You would change the word order and use the fixed introductory word Это. The sentence would become: Это тёплая и красивая шапка. In our current sentence, the adjectives come after the noun because they are the main new information you are sharing about this specific hat.