mention (something)

This means to talk about something briefly. Even if a person says only a few words about a topic, that would be considered "mentioning" that topic. Other words like "say", "talk about", etc. sound like they require a longer period of discussion.

You can use "mention" with a simple noun:

Did she mention me in the article?

Or you can use a dependent clause:

Did he mention where he was going?

You can also use the structure "mention (doing something)"

He mentioned going to the gym earlier in the day.

(question), by any chance?

This is a way to ask a question very hesitantly. Use it when you want to ask a question, even though you're pretty sure that the answer is "no". For example:

I hear a bit of an accent. Are you from Michigan, by any chance?

A: Where'd you go to school? B: Penn State. A: Oh, really? A good friend of mine went there. Do you know Keiko Ishihara, by any chance?

an article

An "article" is a piece of writing that's published in a newspaper or magazine. Sometimes writing that's posted on the Internet can also be called an "article", but sometimes it isn't.

Here are some types of writing that are not called "articles":

something that a person writes on their personal website is usually called a "blog"

a fiction story (a story about something that didn't really happen) is called a "short story".

when the main point of the writing is to review something like a movie or book, it's called a "review"

when a writer tries to explain a personal idea or experience that they have, it's sometimes better to call it an "essay"

Of course, there are many cases where it's OK to call something either a "blog" and an "article", or an "article" and an "essay".