For example, if you had a book of everyone's salaries, you can ask Bob to add $5000 to it. Normally you'd have to give him the key, but with homomorphic encryption you can just perform operations without decrypting it and having Bob know everyone's salaries.
This becomes really powerful for stuff that you need to work with lots of data without knowing the contents of - cloud computing, for example.
From: http://en.wikipedia.org/wiki/Homomorphic_encryption
I don't know much about it either.
For a recent project, I'd be interested in searching and sorting of encrypted data; I wonder if this is up to that. The tricky bit with search is that in many cases, you want to be able to search for substrings, not just an exact match.
http://www.americanscientist.org/libraries/documents/2012861... [PDF]
Try this before Wikipedia.
Basically encryption dchemes that allow operations to be done on the ciphertext that give a result that can be decrypted and is correct plaintext.
Unfortunately, FHE is nowhere near practical enough to do that sort of thing. Maybe in a decade or two we will see FHE implementations used outside the research community.
The number of downvotes my attempt at humour received would suggest that I was wrong.
Please pardon my comment about the trolling. Sometimes it's also hard to determine intentions via text-only communication.