Opportunities abound to conduct research off-campus during the summer. Most summer positions are paid and last approximately 10 weeks. These programs are good way to experience and get to know other universities and their faculty, which can be especially useful if you plan to apply to graduate school in the future.
REUs are the National Science Foundation's summer research programs for undergraduates. As an REU student you are granted a stipend to work with professors or other researchers at a host institution for 10-12 weeks during the summer. Each institution hosts 10 or so REU students. REU application deadlines are in late January or February (the exception is the Cerro Tololo Inter-American Observatory's REU, which takes place during the Chilean summer January-March and so has an early October application deadline).
A comprehensive list of astronomy REUs can be found at:
National Science Foundation's list
For more information about REUs, see:
Berkeley's REU and Summer Research FAQ page
NASA offers paid summer research positions in its locations ranging from Goddard Space Flight Center in Maryland to the Ames Research Center in California. The range of research opportunities is broad, including robotics, space instrumentation, astrobiology, and solar physics. The application deadline is January 23, 2009.
The U.S. Department of Energy sponsors paid research internships at national labs across the country. You can stay close to home at labs such as Lawrence Berkeley and Lawrence Livermore, or you can jet off to labs like Los Alamos and Fermilab. For Fall 2008 internships, apply by August 31, 2008. For Spring 2009 internships, apply by November 30, 2008. For Summer 2009 internships, apply by April 1, 2009.
Many universities, laboratories, and observatories have their own undergraduate research programs. If there's an institution you'd like to work at, odds are they have some sort of program for you! Some useful links which have partial lists of these other opportunities are given below.
The American Astronomical Society's summer employment opportunities
Here you'll find options such as a planetary science internship and a program in the Netherlands.
Space Sciences Laboratory Summer Fellowships
Choose an SSL researcher whose work you're interested in and ask him/her to be your sponsor. Last year's application deadline was May 5, 2008.
Berkeley's REU and
Summer Research FAQ page
A useful overview of REUs and other summer programs.