the separation constant
When you separate a PDE you reach a strange-looking equation: something built only from x equals something built only from t. Think about what that demands. The left side can be changed by moving x while you freeze t; the right side cannot, since it has no x in it. For the two sides to stay equal no matter how you wiggle x and t independently, neither side can actually change at all — both must be locked to a single fixed number. That number is the separation constant.
It is the hinge of the whole method. Substituting u = X(x)T(t) into, say, the heat equation and dividing through gives X''/X = T'/(kT); the only way a function of x equals a function of t is for both to equal a constant, conventionally written -lambda. Setting each side equal to -lambda produces two ordinary differential equations, X'' + lambda X = 0 and T' + k lambda T = 0, that share nothing but the value lambda. The minus sign and the symbol lambda are chosen on purpose: with homogeneous boundary conditions, only DISCRETE positive values lambda = lambda_1, lambda_2, ... will allow a nonzero X, and writing it as -lambda makes those values come out positive.
So lambda is not free — it is quantized by the boundary conditions, exactly like the allowed frequencies of a string. Each admissible lambda_n gives a spatial pattern X_n (an eigenfunction) and a matching time behaviour T_n. That is why the separation constant is also called the eigenvalue: it is the number for which the spatial boundary-value problem has a nontrivial solution. The list of allowed lambda_n is the spectrum, and it sets the menu of modes you superpose.
On 0 < x < L with X(0) = X(L) = 0, the problem X'' + lambda X = 0 has nonzero solutions only when lambda = (n pi / L)^2 for n = 1, 2, 3, ... — these are the separation constants, and the spatial patterns are X_n = sin(n pi x / L). A continuous range of lambda is impossible; the ends snap it to a discrete ladder.
The boundary conditions quantize lambda into a discrete spectrum of allowed values.
Sign conventions vary. Writing the constant as -lambda (so the spatial equation is X'' + lambda X = 0) makes lambda positive for the usual sine/cosine modes; some texts use +lambda. The mathematics is identical — only the bookkeeping of the sign differs.