Conceptually we start at the initial value and march out cautiously along the function. These are much harder than initial value problems, since we must only consider functions which obey the boundary conditions. This ratio is measures by the condition number of the matrix. See NR 16.6, or Stoer and Bullirsch 7.2.16. In an implicit scheme, there is no explicit formula at a points, only a set of simultaneous equations which must be solved to give the solution over the whole grid.
Implicit methods are stable for all step sizes.
In contrast to ODEs, a partial differential equation relates a number of variables and their derivatives. Here we wish to propagate the solution forward in time. The solution is static. Dirichlet conditions specify the boundary values around the region. Von Neumann conditions specify the derivatives at the boundary.