A perfect number is a number which is the sum of its divisors (obviously excluding itself).
Thus, 6 = 1 + 2 + 3 is a perfect number, since 1, 2 and 3 are the numbers which divide 6 evenly. The next perfect number is 28, as 28 = 1 + 2 + 4 + 7 + 14.
Perfect numbers are related to Mersenne Primes (prime numbers that are one less than a power of 2) because a perfect number can be derived from a Mersenne prime. If M is a Mersenne prime, then M*(M+1)/2 is a perfect number. (This was proved by Euclid in the 4th century B.C.) All even perfect numbers are of this form (as proved by Euler in the 18th century). So there is a one-to-one association between even perfect numbers and Mersenne primes.
Only a finite number of Mersenne primes (hence even perfect numbers) are presently known. It is unknown whether there are an infinite number of them. See the entry on Mersenne prime for additional information concerning the search for these numbers.
It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that have helped to locate one or otherwise resolve the question of their existence. It is known that if an odd perfect number does exist, it must be greater than 10160. Also, it must have at least 8 distinct prime factors (and at least 11 if it is not divisible by 3), and it must have at least two prime factors greater than 138.
Some other related information can be found at http://xraysgi.ims.uconn.edu:8080/amicable.html