RSA encryption is a public-key encryption and authentication system. RSA is used in a wide variety of products from such companies as Microsoft, Apple, Sun, and Novell. Both PGP and PEM use RSA encryption.
RSA was invented in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman--hence its name. RSA encryption is a two-key system using both a private and a public key. Anyone can send an encrypted message or verify a signed message using public keys, but only someone in possession of the correct private key can decrypt or sign a message. Encryption and authentication take place without any sharing of private keys: each person uses only public keys and his or her own private key.
RSA works as follows: Take two large primes, p and q, and find their product n = pq; n is called the modulus. Choose a number, e, less than n and relatively prime to (p-1)(q-1), and find its inverse, d, mod (p-1)(q-1), which means that ed = 1 mod (p-1)(q-1); e and d are called the public and private exponents, respectively. The public key is the pair (n,e); the private key is d. The factors p and q must be kept secret, or destroyed.
It is difficult (presumably) to obtain the private key d from the public key (n,e). If you could factor n into p and q, however, you could obtain the private key d. So the entire security of RSA is predicated on the assumption that factoring is difficult; an easy factoring method would "break" RSA.