The largest known prime is the Mersenne prime described above.
The largest known non-Mersenne prime, is 
,
discovered by Brown, Noll, Parady, Smith, Smith, and Zarantonello.
Throughout history, the largest known prime has almost always been a Mersenne prime; the period between Brown et al's discovery in August 1989 and Slowinski & Gage's in March 1992 is one of the few exceptions.
You can help find more primes. For more information see: The Great Internet Mersenne Prime Search home page on http://www.mersenne.org
References
Brown, Noll, Parady, Smith, Smith, and Zarantonello. Letter to the editor. American Mathematical Monthly, vol. 97, 1990, p. 214.