What do groups and rings "look like"? 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?
 A: For Group theory, I can highly recommend "Visual Group Theory" by Nathan Carter. He does a great job delivering a good intuition not only of the concept of a group, but also of different group-related definitions, different groups...
(Like the title suggests; in a visual way)
http://web.bentley.edu/empl/c/ncarter/vgt/gallery.html
For Ring theory, what worked wonders for me is this thread:
https://mathoverflow.net/questions/2748/what-is-the-right-definition-of-a-ring
A: You shouldn't think of a group as a thing in the same way as a vector space is a thing.
Groups are not things, groups act on things.
If $V$ is a vector space, then the collection of invertible linear transformations $V\to V$ is a group.  If $X$ is a set, then the collection of all permutations of $X$ is a group.  If $A$ is any object, in any category, then the automorphisms of $A$?  They're a group.
Now, it turns out that many groups be visualized geometrically.  For example, the collection of rotations in $\mathbb{R}^2$ can be identified with the circle.  So you may from time to time be able to apply your understanding and intuition of geometry, vector spaces, etc., to the theory of groups, but at times you will be very surprised.
Note that every group is the symmetry group of something.  For example, any finite group $G$ is a group of permutations of some finite set, as well as a group of matrices of some finite dimensional vector space.  So it is reasonable to think of any finite group as a collection of symmetries that can be interpreted geometrically.
A: A very basic picture of an algebraic structure is that of just a set with "interconnections" between the elements. Imagine a simple set, an alphabet, say, of two letters, "A" and "B". Now suppose that you can combine the letters to get a bigger set by putting the letters together in any combination (I will talk about a monoid for simplicity). So, something like AAABBBAAB is in this set (the order counts). It's called a free monoid (there is also a so-called identity which is equivalent to an empty combination in our set-up). The "routes" between the words are governed by the rules of our algebraic structure. For instance, start with A, "go" to A to get AA or "go" to B to get B and so on and so forth. In a group, you also have the things like $\text{A}^{-1}$ and $\text{B}^{-1}$ with the rules $\text{A}\text{A}^{-1}=$"empty word" (the same for B). So, you can kinda "walk backwards".
In a ring, you have additional "routes" allowed by the second operation.
But I personally prefer to think of algebraic structures as being like algorithms or computer programs with data and methods. For instance, our monoid above is a program with data consisting of two values A and B along with an identity, say, E, and a method $\text{mult}$ which takes two words and simply concatenates them. This picture does not rely on sets so much, but it is not suitable for all structures of course.
Also, my personal recommendation is to try to think of categories. It gives a much better understanding than that based on set theory.
