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.
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 following summer programs give you an opportunity to encourage younger students to pursue their interest in math.
There are also semester- and year-long math programs for undergraduates. Here are some of the better-known ones: