Pivoting in the word sense means turning or rotating. In the Gauß algorithm it means rotating the rows so that they have a numerically more favorable make-up.
The straight-forward implementation of the LU decomposition has no pivoting. However, it may encounter zeros or near zeros on the diagonal while entries below in the same column have an appreciable size.
So the natural idea is to pick the largest of the remaining entries, call it the pivot (turning axis) and use that row as the basis for the elimination step. To keep constructing the echelon form, rows are swapped or rotated (most efficiently using a row index array), adding permutation steps to the elementary row transformations.
The result of the pivoted Gauß algorithm is a PLU decomposition, where P is a permutation matrix that has in each row and column exactly one entry 1, all other 0.
As to the original matrix, the discretization of minus the second derivative is indeed positive definite. To show that requires an eigenvalue analysis.
For positive definite matrices $A$, the naked LU decomposition without pivoting works, since the diagonal entries that are encountered are quotients of main diagonal minors, and all of them are positive. Symmetry results in $U=D\,L^\top$, so that the $A=LDL^\top$ can be cheaply obtained. If wanted, the square root of $D$ may be distributed to the factors to obtain the Cholesky decomposition.