In many places, including Wikipedia, a homogeneous space is informally described as "a space that looks the same everywhere, as you move through it, with movement given by the action of a group".
I am comfortable with the definition of a homogenous space, which is a space $X$ equipped with a group action of a group $G$ such that $G$ acts on $X$ transitively, but I am struggling to create a mental picture of what this definition represents. How does this definition lead to saying "a homogeneous spaces looks the same everywhere"?
Best Answer
Two nice examples to visualize are a sphere $S^2$ or a plane $\mathbb{R}^2$, which are homogeneous due to the action of the group $SO(3)$ of rotations and the group $\mathbb{R}^2$ of translations respectively. The sense in which these spaces "look the same everywhere" is that if someone plopped you down at a point in the space and asked you to figure out where you were you wouldn't be able to do it, because no point in the space has any "local features" which distinguish it from any other point.
More generally, if $X$ is a space, we might say that two points $x, y \in X$ "look the same" if there is a homeomorphism (or diffeomorphism, etc. depending on your purposes) $f : X \to Y$ such that $f(x) = y$. This is an equivalence relation on points in $X$ and $X$ is homogeneous iff every point is equivalent under this equivalence relation.
For a non-homogeneous example you can consider the closed ball $B^2$ as José suggests in the comments. Here there are two types of points, points in the interior and points on the boundary (two equivalence classes of the "looks the same" relation), and they can be distinguished by the behavior of their open neighborhoods. Another easy example to visualize is something like a star graph, considered as a topological space. Here there are three types of points: the outer vertices, the points in the interior of the edges, and the central vertex, and they can again be distinguished by their open neighborhoods.