As far as the attempt to describe mathematics research to others:
Doing mathematics is like helping to build a beautiful house. I am drawn to help because I have a love for what the finished house will be, and beyond this, because I enjoy the challenge of building. But my role is small. It may only consist of choosing the right location for the foundations, or determining how much insulation to put within the walls, or finding the right kind of wood for the exterior. If I speak with someone who admires architecture, but has never built a house themselves, I worry that focusing on technical issues will blunt their interest in what I am doing, or rather, obscure the larger picture.
Yet the analogy is not absolute, because I find that the technical issues of mathematics are inevitable in a way that I do not believe specific choices about location, insulation, wood, etc. are always inevitable. Building a real-life house leaves a great amount of room for chance, arbitrariness, non-necessity. Maybe it is more revealing to say that the mathematical technicalities that have interested me have held a naturalness that inheres without my needing to keep in mind some vaster project. However, I cannot even describe them without first explaining a good deal of the rest of that vastness.
In the end, I usually resort to showing sketches of what the house will look like from afar, where the details become blurry. But these sketches convey nothing of how it feels when one is building — placing one’s fingers to the grain of the wood. I do not want people to believe that drawing a picture is the same as building, though the former may well be a step of the latter. The question is how to bring you a small piece, or perhaps a model, that you can touch, touch every way.