One way to think of a topos is as some kind of fancier category of sets.
One way to make sets fancier is to consider sheaves of sets on some topological space; this is mentioned in Zhen Lin's answer. One can think of a sheaf of sets as a set which varies and twists over the topological space, but it seems that this could be a bit painful for you to think about with the background you're coming from.
Another way to makes sets fancier is to put actions on them. So:
Let $G$ be a finite group, and consider the category of all finite $G$-sets,
i.e. all finite sets equipped with an action of the group $G$. (Morphisms are maps between sets that are compatible with the $G$-action on source and target.)
This is an example of a topos, which is pretty small in your non-technical sense.
The subobject classifier is the two element set $\Omega$, with trivial $G$-action, and with
one of the two points distinguished. If $X$ is a $G$-set, and $Y$ a $G$-invariant subset, then we have the morphism $\chi_Y: X \to \Omega$ which maps
all of $Y$ to the distinguished point in $\Omega$, and all of $X\setminus Y$ to the other point. This morphism "classifies" the subset $Y$. (More precisely, $Y$ is the preimage of the distinguished point of $\Omega$ under $\chi_Y$.)
If we take $G$ to be the trivial group, then we just recover the topos of finite sets.
Let's instead take $G$ to be the cyclic group of order two, say $G = \langle 1,\tau\rangle,$ with $\tau$ of order two. Then to give a finite $G$-set is just to give a finite set $X$ equipped with an involution (i.e. permutation of order two) $\tau$.
Now in addition to the two-element set $\Omega$ with trivial $G$-action, which (once you designate one of its points as being the distinguished one) is the subobject classifier, you could think about the two-element set $\Omega'$ equipped with the non-trivial involution, which switches the two points.
We have already seen that $Hom_G(X,\Omega)$ (I write $Hom_G$ for maps preserving the $G$-action) is equal to the collection of $G$-invariant subsets of $X$.
What about $Hom_G(X,\Omega')$? You could try to compute this (of course it will depend on the particular $G$-set $X$). It's not particularly exciting, but it might help you get a feeling for the difference between the subobject classifier and some other objects, such as $\Omega'$.