What are the types of pseudopotentials?

To add to Nike's list, one should also differentiate between the energy consistent pseudopotentials used in quantum chemistry and the shape consistent pseudopotentials that dominate in the solid state community. Energy consistent pseudopotentials are also shape consistent, while shape consistent pseudopotentials result in much larger errors in the energy than with the energy consistent potentials.

This minireview by Peter Schwerdtfeger might also be useful: ChemPhysChem 12, 3143 (2011).

Summary of the "milestone" pseudopotential (PP) papers

Since it wasn't available anywhere & took me a few hours, it's an answer rather than question-edit:

Local pseudo-potentials:

• 1935: Zusatzpotential / Hellmann (Generally credited as the first pseudopotential).
• 1936: Fermi pseudopotential (for s-wave scattering of a free neutron by a nucleus).
• 1958: Harrison (FPPM: First-Principle PP method, fitted to nearly free e- Fermi surface of $ce{Al}$).
• 1959: Phillips-Kleinman (core-val. orthogonalization terms replaced by "hard-core" potential).
• 1968: Weeks-Rice (extended the single valence e- work of Phillips to many valence e-).
• 1973: Appelbaum-Hamann potential (smooth potential for $ce{Si^{4+}}$ $rightarrow$ works for band gaps in $ce{Si}$).

Non-local pseudo-potentials:

• 1979: HSC (norm-conserving PP: exact energies & nodeless $psi$ for $r>r_c$).
• 1980: Kerker PP (non-singluar PP: exact for $r>r_c$, $dotpsi$ and $ddotpsi$ matched with ab initio).
• 1982: Kleinman-Bylander (highly simple).
• 1990: RRKJ Optimized PPs (improve plane-wave convergence).
• 1990: Ultra-soft / Vanderbilt (norm-conservation not needed for $psi$: allows far more flexibility).
• 1990: Troullier-Martins (scheme for generating very soft norm-conserving PPs for PWs).

Generalizations of pseudo-potentials:

• 1994: Blöchl (generalization of PPs and LAPWs based on a linear transformation).