Nonstandard Logics and Extensions

Computability Logic
Wikipedia (free encyclopedia) article.
http://en.wikipedia.org/wiki/Computability_logic

Infinitary Logic
Article in the Stanford Encyclopaedia of Philosophy by John L. Bell. Infinitary Logic is a branch of formal logic where finitary formulae are replaced by potentially infinitary mathematical entities.
http://plato.stanford.edu/entries/logic-infinitary/

Logic System Interrelationships
Shows how a number of representative logics fit together. The interrelationships usually given as something of the form: System X is system Y plus the axiom Z. By John Halleck.
http://www.cc.utah.edu/~nahaj/logic/structures/

Non Standard Logics
A comprehensive listing of flavours of non-standard logic with brief descriptions and references, compiled by Peter Suber.
http://www.earlham.edu/~peters/courses/logsys/nonstbib.htm

What are Weak Arithmetics
Notes defining the subject. Available in HTML and PS formats.
http://www.univ-paris12.fr/lacl/jaf/html/wa.html