Here are some examples:
* Correct: I found a beautiful agate at the beach.
* Correct: The Agate Creek in Montana is known for its unique gemstones.
* Correct: The "Agate of Fire" is a famous gem in the museum.
However, if you are referring to a specific company or organization named "Agate," then you would capitalize it.
* Correct: I bought my new computer from Agate Technologies.