Homology as Boundary of "Submanifold" In the plane, imagine a horizonal figure eight, $\infty$. Let $\alpha$ be the curve which is convex from the leftmost point of the figure to its "middle", and concave from the middle until the rightmost point. Let $\beta$ be its complement (the remainder of the figure eight). Orient both $\alpha$ and $\beta$ from left to right. These two curves are homotopic, so they must be homologous, the latter true iff their difference is homologous to zero. From what I understand, geometrically, being homologous to zero means being a boundary of some "submanifold".
What set in the plane has the figure eight as its boundary?
Edit:
I changed the title to fit the question better.
 A: The quotes on the "submanifold" are important. It's hard to visualize this case, because the curves $\alpha$ and $\beta$ intersect.
When the curves don't intersect then things are a bit easier to see. For example, consider the $2$-dimensional torus. Two disjoint longitudes on the torus are homologous, and this is geometrically justified because the meridians bound a $2$-dimensional cylinder embedded in the torus.
When the curves intersect, you have to use the fact that the relation "$\alpha$ is homologous to $\beta$" is an equivalence relation. So in the figure-eight case, find an oriented curve $\gamma$ that has the same endpoints as $\alpha$ and $\beta$ and are homologous to both $\alpha$ and $\beta$ without intersecting them except at the endpoints. Then $\alpha - \gamma$ bounds a submanifold $U$ of the plane that is equivalent to the open disk, and $\beta - \gamma$ bounds another submanifold $V$ that is equivalent to the open disk. Then in some sense, $\alpha - \beta = (\alpha - \gamma) - (\beta - \gamma)$ bounds the "difference" of the two submanifolds, i.e. $\alpha - \beta$ bounds $U - V$.
The "difference" of two submanifolds doesn't have any geometric meaning, it's a formal difference that arises from the algebraification of the idea of submanifolds and boundaries implicit in homology. Hence why I said the quotes on "submanifold" are important.
