Mathematics

• Math StackExchange, a Q&A forum for math of any level (homework problems welcome, if you explain what you've tried so far)

• MIT's OpenCourseWare

• Tricki, a wiki-style collection of mathematicians' favorite problem-solving tricks and techniques

• How to Solve It, by George Pólya

• 10 Ways To Think Like A Mathematician, by Kevin Houston

• Math Reference Project, maintained by Karl Dahlke 

• MathWorld

• A Mathematician's Lament, by Paul Lockhart

TeX / LaTeX

The document preparation system LaTeX is almost universally used to create mathematical and scientific works. Here is an explanation of what TeX and LaTeX are, their history, and the benefits of using them. 

• LaTeX Wikibook

• TeX StackExchange, a Q&A forum for all things TeX- and LaTeX-related

• TeX Resources on the Web, organized by the TeX Users Group

• LaTeXiT, a LaTeX equation editor

• Detexify, a website to help you find the LaTeX symbol you're looking for 

• BibTeX and bibliography styles 

• TeXnicCenter, a Windows LaTeX editor

• TeXShop and iTeXMac, LaTeX editors for Mac OS X

• Kile, a LaTeX editor for KDE 

• Texmaker and TeXworks, LaTeX editors supporting Windows, Mac, and Linux 

• GmailTeX, a plugin for Gmail that compiles TeX / LaTeX code in your emails
   

Computation

• Wolfram Alpha

• List of software that is available using a license from Brown (includes Mathematica, Maple, and Matlab)

• Mathematica StackExchange, a Q&A forum for Mathematica

• Sage Notebook

• PARI/GP
   

Programs for Undergraduates

Brown offers Undergraduate Teaching and Research Awards (UTRAs). Applications for summer UTRAs are typically due in early February, so you should definitely talk with the professor you'd like to work with before winter break.

Outside of Brown, many summer math programs are what are known as "Research Experiences for Undergraduates" (REUs). These are designed to give you a sense of what real mathematics research is like (though the specific format can vary greatly). Many programs will require one or more recommendation letters, and it is good to give professors some time to write these.

• The NSF's list of math REUs

• The AMS's list of math REUs

• The AMS's list of math internships

• Some advice from the MAA: Is An REU For You?

• Summer@ICERM

• UC Berkeley's Geometry, Topology, and Operator Algebras REU

• The TMD (Turkish Mathematical Society) Undergraduate and Graduate Summer School

• International Summer School of Mathematics for Young Students

• Park City Mathematics Institute

• The SMALL REU at Williams College

• Claremont Colleges Mathematics REU

• Duluth Undergraduate Research Program

• NSA Director's Summer Program (DSP)

• Research in Industrial Projects for Students (RIPS)

• Summer Math Institute at Cornell University

• Summer Program for Women in Mathematics

The following summer programs give you an opportunity to encourage younger students to pursue their interest in math.

• PROMYS, an excellent program for high schoolers at Boston University, focusing on number theory

• The Ross Mathematics Program, held at Ohio State University (PROMYS is a descendent of the Ross Program)

• HCSSiM, a program for high schoolers held at Hampshire College

• Center for Talented Youth, a network of summer programs run by Johns Hopkins University

There are also semester- and year-long math programs for undergraduates. Here are some of the better-known ones:

• Mathematics Advanced Study Semester (MASS) at Penn State

• Budapest Semesters in Mathematics

• Math in Moscow (see this scholarship offered by the AMS)

• Junior Program at Smith College's Center for Women in Mathematics

Applying to Graduate School

• A mathematician's survival guide: graduate school and early career development, by Steven Krantz

• Career advice, by Terence Tao

• Some good fellowships to apply for:
  • NSF Graduate Research Fellowship Program
  • NDSEG Fellowship 
  • Hertz Foundation Fellowship
  • NSA Graduate Fellowship
  • The Churchill Scholarship 
  • Ford Foundation Fellowship

• Some advice on how to apply for those fellowships:
  • Applying for Graduate Research Fellowships, by Seth Sullivant
  • Advice for applying for graduate science fellowships: NSF, NDSEG, Hertz, by Philip Guo
  • Fellowships and Funding advice from Cornell Astronomy (most of it is relevant to all subjects)
  • Advice on applying for the NSF fellowship, by Kevin Sanft

• Some graduate school links on the AMS's website.

• The official page for the GRE Math Subject Test.

• Assistantships and Graduate Fellowships in the Mathematical Sciences, a document compiled by the AMS each year

• Preparing For Grad School, advice from MIT

• Some grad school advice from Harvard

• Some grad school advice from Haverford.

• How to get a great letter of recommendation, by Matthew Might

• Recommendations, by Ravi Vakil

• Get into grad school for science, engineering, math and computer science, by Matthew Might

• Thoughts on graduate school, by Noah Snyder (an addendum by Ben Webster)

• More advice for prospective graduate students, by Noah Snyder 

• A Graduate School Primer (MAA), by Doug Faires 

• Advice on picking a Ph.D. program and advisor 

• Choosing a graduate school, by Diana Davis

• Tips for Visiting Grad Schools, by Kimball Martin 

Organizations

• American Mathematical Society

• Association for Women in Mathematics

• Mathematical Association of America

• Society for Industrial and Applied Mathematics

• Young Mathematicians' Network (their site is currently under construction)

Miscellaneous

• Obvious But Underappreciated Truths, by Alan Stein

• The History of Mathematical Symbols, by Douglas Weaver

• The MacTutor History of Mathematics archive

• A neat video (part 1 and part 2) giving an excellent explanation of the sphere eversion.

• An inspiring documentary about the proof of Fermat's Last Theorem

• Möbius Transformations Revealed

• Science Lives Project,  a series of interviews with famous mathematicians and scientists