Last November, a new largest prime was discovered, a
Mersenne prime in particular: 2^20996011 - 1.  It is the 40th known Mersenne prime, and was discovered as part of a team effort known as the GIMPS Project (Great Internet Mersenne Prime Search).  Over 200,000 computers are running the
Lucas-Lehmer test (a special number-theoretic test for
Mersenne numbers to test primality) on Mersenne numbers; the project has been growing in scale since 1996.  (A Mersenne number has the form 2^p - 1 where p is prime.)

   For more about the GIMPS Project, link to

http://www.mersenne.org/

where you may find downloadable code to run the Lucas-Lehmer test on a range of exponents given to you when you join up with GIMPS.  Link to the"status" page within this website to see the ranges of exponents that have
been completely tested, and where there is still much work
to be done. This makes for a nice project for a school's math club, to get students together to form a team to track their progress, and learn something about number theory and primality testing in the process.

   For more about primes and primality testing in general,
please visit Chris Caldwell's primes page at the University of
Tennessee-Martin:

http://www.utm.edu/research/primes/

Here, you will find details not only on Mersenne primes,
but other special types of primes: Sophie Germain, primorial, p*2^q - 1, factorial, Generalized Fermat, Cullen, Woodall, and others.  There is quite a wealth of information at this site - a must for anyone interested in knowing more about prime numbers, their underlying theory, and further resources on
primes.