The ability of the RSA cipher (Dave's note: what's a cipher? We need a link to the glossary here.) to protect electronic communication lies in the separation of the encryption and decryption process. To encrypt something using the RSA scheme, a person needs the recipient's public key. This is the parameter used to generate a ciphertext. Each party has his private key; this is used to decrypt messages that were encrypted with the corresponding public key.
A public key is given out freely; a private key, on the other hand, is confidential.
Suppose a hacker is lucky enough to have caught you and another person exchanging public keys; without knowing the private keys, they cannot read your online communications. Such a hacker could send encrypted messages to you, but could not read messages sent to you.
There are a few weaknesses a hacker could exploit (particularly with the ciphertexts and public keys) which would give access to your communication. At EngFrosh, however, selection of our public key and private key have minimized the effectiveness of such cryptoanalysis.
If you are interested in RSA and the EngFrosh Registration Applet, you are encouraged email the author, Jean-Luc Cooke.