Weyl Semimetal and Dirac semimetal

Here are a few thoughts:

Weyl semimetal

Weyl point. A Weyl point is a point at which 2 bands cross. This places severe constraints on which type of material can host Weyl points, because in materials with both time-reversal and inversion symmetries, bands are already doubly-degenerate (spin up and down have the same energy at every $mathbf{k}$-point), so all band crossings will involve 4 (rather than 2) bands and will therefore not be Weyl points. This means that a prerequisite to have a Weyl point is that the material breaks time reversal symmetry (e.g. ferromagnet), or inversion symmetry. The latter futher requires that there is a spin-dependent term in the Hamiltonian to actually split the spin degeneracy of the bands (as inversion symmetry breaking alone is not spin dependent), which can be accomplished with spin-orbit coupling.

Weyl semimetal. A Weyl semimetal is simply a material that has a Weyl point at (or very near) the Fermi energy.

Characterization. A Weyl point is a source or sink (monopole) of Berry curvature, so you can characterize a Weyl point by constructing a surface in $mathbf{k}$-space that encloses the Weyl point, and then calculating the Chern number over that surface. It will be $+1$ or $-1$ for Weyl points, and the sign is called the chirality of the Weyl point.

Codes. Most codes that allow you to calculate topological properties can be used to characterize Weyl points. For example, Z2Pack or WannierTools, both of which have interfaces with some of the major DFT codes.

Dirac semimetal

A Dirac semimetal is similar to a Weyl semimetal, but now the crossing involves four bands rather than two. This means that the symmetry breaking conditions are not as severe, and a well-known example of a material that hosts Dirac points is graphene. In fact, the very presence of Dirac points in graphene is protected by time-reversal and inversion symmetry (see here).