Do you tell the people who ask you this question that you're a writer? If not, take a few minutes to think about why. What would it take to make you a "real" writer? There are many common reasons why even highly skilled writers are afraid to claim the title.
"I don't have a degree in English or creative writing."
Neither did Mark Twain, but that didn't stop him from writing some of the cleverest books in the English language. There's nothing wrong with studying writing, of course. If can help you learn new techniques and give you a basic knowledge of different styles and traditions. It is not necessary, though, to obtain a degree before calling yourself a writer.
"I don't make my living as a writer."
Which are you the most passionate about, your writing or your day job? If you love them both, then claim them both. ("I'm a freelance writer and a hospice social worker.") Otherwise, simply tell people the truth that is in your heart. You are a writer.
"I haven't published anything yet."
Wait a minute. Are you sure you haven't published anything? What about an editorial in your local newspaper, an article for the PTA newsletter, or a blog post? Any of these things--and others like them--count as publications. And anyway, you don't have to be published to be a writer. As long as you have written something, the title is yours.
"I've never won a writing award."
It's nice to have your talent recognized by others, but you can't let your identity depend on the opinion of a panel of judges.
"People will ask me what I've written."
Some people assume that every writer pens novels, short stories or at the very least, magazine articles. If those items aren't your niche, just explain what you do write. ("I ghost write books on senior care." or "I write blog posts about sales and marketing.")
By now I suspect you're getting the idea. A writer is a person who writes. I was a writer from the moment I first scribbled a two- or three-sentence short story in first grade. I was a writer long before my first piece of work sold. (I was in high school. I got paid a few bucks for submitting a blooper I'd caught on my favorite soap opera.) I'm a writer now, and I fully expect to be a writer for the rest of my life.
What do you tell people when they ask what you do?