How to explain your area of study to non-math people I'm sorry in advance if here is not the suitable place to ask this question, and people can feel free to vote to close this if that's the case. However, since I'm not sure about this issue, I'll ask it in any way:

I was reading Hungerford's Algebra, and my friend working on food science saw the title "Fundamental Theorem of Galois Theory", and asked me what it's about. (Seeing the word "fundamental", he thought it should be a very basic, simple concept.) Then, I said something like "there are structures called groups, fields, and they have some special subsets called subgroup, subfields. In order to understand subfields, mathematicans makes a correspondence between subgroups of some associated group, which is easier to work with and better understood, compared to fields." 

He said it made some sense, but I'm sure it was a terribly bad explanation. My question is how do people generally deal with this issue? Is this only my problem (which really might be the case), or is this a general problem with pure math people?

This issue really annoys me in the sense that, for example, when you consider some person working on food science, psychology, chemistry etc., they're generally able to tell roughly what they're trying to do. But, I'm always having very hard time when I'm asked the same question. I'm a PhD student now, and I plan on spending whole my life with mathematics. But, I'm afraid that, say in my 50 (if I live that much), if someone asks me what I did in my entire 50 years, and I fail to answer, that would be a shameful situation for me, I believe.

I tried my best to express my question in a good way, but still I'm sorry if it's not good enough, and don't hesitate to let me know if I should write some parts more clearly/explicitly.
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.
My six pence:

Tell them that you are studying how the four elementary operations ($+,-,\times,\div$) generalize to other things than ordinary numbers. For instance, when you are limited to a range like $[0,255]$. It turns out that this theory is extremely rich and powerful, and we are still far from understanding all of it. I has numerous real-world applications like cryptology (avoiding that your data be stolen on the cloud) or error correction (ensuring that your scratched DVD is still readable).