Here I will list various useful books, websites, videos and online PDFs that are useful for learning molecular computational chemistry.
If you are new to computational chemistry, don't try to learn the subject by reading research papers. That will only get you confused. Reviews on computational chemistry are better but it's best to find real educational material like a textbook or online educational material.
Recommended books
Start with one of these, they are nice overviews of most methods and theory:
Essential of Computational Chemistry - Christopher J. Cramer
Introduction to Computational Chemistry - Frank Jensen
Then look at these:
A Chemist's Guide to Density Functional Theory - Koch and Holthausen
Molecular quantum mechanics - Atkins
Modern Quantum Chemistry - Szabo & Ostlund
The ABC of DFT by Kieron Burke (free online book on DFT)
Websites
Jan H Jensen's brief introduction to basis sets
Nice courses on computational chemistry with good online material:
http://www.cup.uni-muenchen.de/oc/zipse/computationalchemistry1.html
http://www.cup.uni-muenchen.de/oc/zipse/contents-table-of-contents.html
http://www.cup.uni-muenchen.de/oc/zipse/computationalchemistry2.html
Excellent website for the Minnesota Computational Chemistry course by Prof. Chris Cramer. Includes PDFs and lecture videos. See also Cramer's Youtube page.
Kieron Burke lectures on DFT (videos)
Lecture series in Electronic Structure Theory (David Sherrill and others)
ORCA Summerschool 2014 Youtube videos
ORCA Input Library: Useful resource for running ORCA calculations (maintained by RB)
Interactive molecular dynamics:
HTML5/Javascript interactive simulation
Useful educational papers and reviews
Basis sets
Review: Gaussian basis sets for molecular applications by Grant Hill. Int J. Quantum Chem
Review: Atomic orbital basis sets by Frank Jensen. WIRES Comp. Mol. Sci.
Thermochemistry
Entropy Explained: The Origin of Some Simple Trends
Solid state
How Chemistry and Physics Meet in the Solid State - Roald Hoffmann
A chemical and theoretical way to look at bonding on surfaces - Roald Hoffmann
Blogs and other sites
Molecular Modeling Basics - Nice blog by Jan H. Jensen, accompanying his book. See also his Youtube page.
Chemical Quantum Images - Felix Plasser. Nice posts on general theory, excited states, visualization tools, useful scripts etc.