Taking undergraduate physics courses, I had to deal with Euclidean vectors often. In classes like Calc III, the concept was also there.

I'm not sure if this is why, but I've always had a more intuitive "picture" of what a vector space was than other algebraic structures. Even though in a linear algebra course, vector spaces are as arbitrary of a structure as any other, this association with a "space of scalable directed lines" stuck. It makes the concept of "dimension" of a vector space very intuitive, along with many other things.

For rings and groups, and other structures, I have no such intuition. I've heard groups compared to all sorts of things, involving symmetries, and Christmas tree ornaments. I don't see these things. I have completed graduate courses on group theory and am currently self-studying rings, but have little intuition on these things.

In other words, if I had to explain a vector space to someone with no knowledge in mathematics, I would probably go the route of explaining the three dimensional space with scalable directed lines, a very concrete example, and could do so in plain language comfortably and intuitively. If I had to explain a group, I would really have no choice but to say "a group is a set of objects endowed with a binary operation such that..."

What is your intuitive notion of these other algebraic structures? How do you "visualize" a group?