I have to answer both of these questions alot. I've found most people are more interested in the sensation than the explanation.
I use tea to describe tannins. Explain it as the feeling you get on your gums from tea. Explain that's why acid is so important to wine to provide a way to moderate or balance that feeling.
Most people actually use 'dry' not in its proper definition of having no residual sugar, but in reference to mouth feel. You can explain how the term should be used, but most people are actually talking about the effects of tannins when they say 'dry'. They are talking about the drying effects of tannin.
So if someone says they like a dry wine, they like big tannins. If they say a wine is 'too dry' which I hear a lot, they mean they like less tannin or better tannin/acid balance. People who know what dry actually means in wine terms rarely use the word to describe a wine, so you're probably pretty safe assuming that when some says they like a dry wine, they are not talking about sugar.