Breakdown of Эта шляпа очень старая.
старый
old
этот
this [f: эта, n: это]
очень
very
шляпа
hat (with a brim)
Questions & Answers about Эта шляпа очень старая.
Why do we use Эта here instead of Это?
Because Эта is acting as a demonstrative word attached directly to the feminine noun шляпа to mean "This hat". If you were just stating "This is a hat", you would use the fixed, gender-neutral word Это regardless of the noun's gender, as in Это шляпа.
How would the sentence change if I wanted to say "This is a very old hat"?
You would switch back to the neutral pointing word Это instead of the feminine Эта. The sentence would become Это очень старая шляпа. In that structure, Это acts as the phrase "This is", rather than pointing at "This specific hat".
Where is the word for "is" in this sentence?
In Russian, the present tense of the verb "to be" is almost always skipped. You literally just say "This hat very old". You only need to use a "to be" verb if you are talking about the past tense (была) or the future tense (будет).
Why does the adjective старая end in -ая?
Adjectives must match the gender, number, and case of the noun they describe. Because шляпа is a feminine noun in the nominative case (meaning it is the subject of the sentence), the adjective takes the standard feminine nominative ending -ая.