Kernels are used for distance measurement in a higher dimensional feature space. In this way, a transformation in this feature space can be simulated ("kernel trick") and discriminant function can be learned which are linear in the higher dimensional feature space but non-linear in the original feature space.
The precomputed kernel (matrices) used in LibSVM are the result of applying a kernel function and contain the (kernel-) distances between all training (and test) instances.
If only the data and the kernel to apply are used as parameter in the LibSVM call, the kernel matrice is computed by LibSVM itself (possibly more efficient).