The slashdot editor added: The Wikipedia entry for homomorphic encryption calls it "an extension of either symmetric-key or public-key cryptography."
to the end of the post as if that somehow explains to people what homomorphic encryption is.
Homomorphic encryption is any encryption scheme that allows you to perform computation on the encrypted data without decrypting it. So, for example, if you had heart rate data that was decrypted and stored on your phone, you could send it safely fully encrypted to a web service that could then calculate the average and send it back to you to compute your average HR. But the web-service itself wouldn't have any idea what your average heart rate is, it would only know an encrypted version of your average heart rate, which it would send back to your device. Your device would decrypt it and show it to you.
This is different than normal encryption schemes where the encrypted version of the data is essentially useless and must be decrypted to compute anything with, so if you want a 3rd party to compute something on your data, you have to trust them to work with the decrypted data.