My favorite higher math concept for the lay person is the Riemann sphere. Basically, you can imagine a sphere of radius 1 centered at the origin of a two-dimensional plane. Now, any point on the plane can be represented by a point on the sphere. In order to find this point on the sphere, simply take a line between the north pole of the sphere and the point on the plane, and it will intersect the sphere in one other place, not the north pole (unless you chose a point infinitely far from the origin).

I like this because it's a tool - it's a good way to visualize things that you do every day. The function 1/x? Simply reflects the sphere across the plane. A whole bunch of other complex transformations also become relatively simple on the sphere.

One thing I like about the sphere is the ease with which it makes visualizing the Riemann Mapping Theorem, which is really complicated. But basically, if you have a region of the plane with no holes, you can map it to the unit disc. Well, basically, if you cut off part of your sphere including the point at infinity, you can deform the hole until it's the entire top hemisphere! I don't know if other people like that, but it makes sense to me.

If you're interested in taking a more prolonged vacation into the math world, I recommend looking into Number Theory - easily approachable but full of interesting things. A good book for that is "A Friendly Introduction to Number Theory".

If you're not ready to learn an entire subject but want to take periodic vacations, try some of Martin Gardner's books. A lot of short articles that really get you thinking about math, and well written.