the heat initial-boundary-value problem

To predict the temperature in a metal bar you need three things: the law it obeys, what it looked like at the start, and what is happening at its two ends. Give the heat equation alone and you have not said enough; the heat initial-boundary-value problem is the complete, well-posed package of equation plus initial condition plus boundary conditions that pins down exactly one temperature history.

On an interval 0 < x < L for times t > 0, the problem is: solve u_t = k u_xx, with an initial condition u(x,0) = f(x) telling you the starting temperature everywhere, and a boundary condition at each end x = 0 and x = L for all time. The boundary condition can be of three standard kinds. Dirichlet fixes the temperature itself, e.g. u(0,t) = 0 (an end held in an ice bath). Neumann fixes the heat flux, i.e. the spatial derivative, e.g. u_x(0,t) = 0 (an insulated end, no heat in or out). Robin mixes them, u_x + h u = g (an end losing heat to surroundings by Newton's law of cooling). You need one such condition at each end, plus the one initial condition, because the equation is first order in time and second order in space.

Why exactly this much data? It is what makes the problem well-posed in Hadamard's sense: a solution exists, is unique, and depends continuously on the data. The maximum principle is what guarantees uniqueness and stability here. Get the data count wrong and the problem misbehaves — too little and it is underdetermined, and notably running it backward in time (the backward heat equation) is ill-posed no matter how you pose the data. Separation of variables solves this problem cleanly when the boundary conditions are homogeneous (zero) and constant-coefficient; otherwise you first shift to a steady state or use Duhamel's principle for sources.