I got that question within my first year going veg. My mom and I were eating at a Brazilian restaurant and I turned down roast beef that a waiter was slicing up on a portable cart. I think I told her I was trying to eat healthier. I was not ashamed of being vegetarian. I had a real reason to hide it from her. Before I was born, my grandmother died from malnutrition because she was vegetarian. So, of course when my mom found out I went vegetarian, she flipped out and lectured to no end. To ease her mind, I researched nutrition like crazy and told her that her mother probably died because she didn't know enough about nutrition and wasn't eating right. Today, the internet makes it so much easier. She researched into it, too. I think she learned that being vegetarian is actually a healthy lifestyle if you do it right. It's kind of funny, once in a while she will run across something and tell me about it. The only thing she got one up on me is the fact that not all nutritional yeast contains B12. Every other nutritional fact I already knew.
Do your research and have facts to back yourself up. You will most likely get some resistance. Most people hate change. They will worry about your health, how much trouble it will cause them with cooking for you, they fear what they do not know much about, etc.