The origin is at the point where the $x$, $y$, and $z$-axes meet, as usual.
The numbers in parentheses are Miller Indices (as Mark pointed out in his answer). In the case of a simple cubic crystal, the indices $(h k l)$ represent a plane that passes through the points $(1/h,0,0)$, $(0,1/k,0)$ and $(0,0,1/l)$. More generally, the indices are the coefficients of a vector orthogonal to the plane, in the basis of primitive reciprocal lattice vectors (you can check that these two definitions are equivalent for a cubic crystal).
In the example figure on the right, the Miller indices are $(1 \bar{1} 1)$, so the plane should intersect the $x$, $y$, and $z$-axes at $1$, $-1$, and $1$ respectively. The $x$- and $z$-intercepts are explicitly shown in the figure, but the $y$-intercept is not; it would be visible if the figure was extended to the left by one unit.
Similarly, for the example figure on the left, the plane will intersect the $x$, $y$, and $z$-axes at $-1$, $1$, and $1$ respectively. Here the $y$- and $z$-intercepts are shown; the $x$-intercept would be visible if the figure was extended back into the screen by one unit. Notice the symmetry between the two cases.
The miller indice is the inverse of the distance between the planes, in the direction of that indice. The higher the index, the more planes in the unit volume.
In the case of $(1,1,1)$, $n=1$,$k=1$ and $l=1$.
Best Answer
I could only find this poor-quality picture. It should give you an idea, anyway. For example, consider the first picture in the first row: $(l,m,n)=(1,0,0)$ in that case, and it is easy to verify that the distance between the grey planes is
$$d=a$$
In the second case, $(l,m,n)=(1,1,0)$, and you can see that
$$d=\frac a {\sqrt 2}$$
etc.