How to explain your area of study to non-math people [duplicate]

The difficult task of explaining what you do or study as a mathematician in layman terms is a common experience.

I think that the main mistake is to try to be precise nonetheless. In your case of Galois theory, terms as "fields" or "groups" are useless and may even lead to more confusion (I remember once a physicist getting really confused at a talk because he had never been exposed to the algebraic meaning of the term "field"). I would have said something along the following lines:

"There is a large class of numbers called algebraic numbers which need to be considered when solving simple equations in just one unknown. They include most numbers people are familiar with, like usual fractions or their roots, but don't include others, like $\pi$. The set of algebraic numbers has symmetries, which are in some sense like spatial symmetries, but less easy to visualize. Galois theory is the theory of these symmetries."

From here one could move on (for instance telling how the $\pm$ sign in the formula for the quadratic equations--something that everybody has seen in his/her school years--is a manifestation of these symmetries) according to the interest shown by the person we are talking to.

I am far from being a mathematician, but I deal in other areas of abstract thinking. My model is that all knowledge is based on layers... in math terms we start with numbers, then we add operators, then jump to algebra, then on to calculus... From the above, I surmise that Galois Theory is a few layers below (above) that. [A layman's view!]

The problem comes in when you are trying to explain concepts that are beyond one or two layers from what the person is comfortable with - such as explaining vector calculus to someone who can only do arithmetic. Or routing protocols to a smartphone (only) user. Or the biochemistry of cancer to anyone who thinks you catch a cold by being cold (it's a rhinovirus infection people!).

In these cases, I'll aim to take them to the next level and then say this is an example of {Galois theory, Internet technology, cancer research,...}. This builds on what they know already and usually provides a workable answer to the initial question without confusing or being meaningless words. Then if they ask more questions, they are usually being more specific and you can give more focused questions.

2c (& not worth a penny more).