Imagine you're a head chef at a restaurant, and a panel of food critics has come in. You have a kitchen staff who range from the experienced to the newbie.
Every plate that gets sent out carries your reputation with it. Wouldn't you want to have it as inspected, checked and tasted as possible before it reaches your diners?
Most of the people who ask me, although they don't work in IT, interact with software on a daily basis. Normally I just ask them if they have found a bug in any websites or software packages they use. I then explain that I help to prevent bugs like that making it into the finished product.
complete technophobes (e.g., my mom) usually don't ask too many questions - so I just say something like "I make sure everything works the way it should".
for people who do use computers but don't exactly understand what goes on behind the scenes, I usually give examples of bugs found in applications almost everybody uses - like facebook chat.
Always a struggle. I think that the best way to do it is to explain a context that is familiar to them. If they use internet banking as most do these days, thats a good subject as most people think they understand it.
I wrote a blog recently on this subject that may help try this
It's a hard job trying to explain non-testers (and managers) why we do (or want to do) things like we do, therefore I have switched over and started to draw cartoons about software testing at www.zelger.org
Everyone understands those pictures even if you don't know all the details.
You might want to read http://www.satisfice.com/articles/explaining.shtml by James Bach :)