Which cloud service are you referring to? You posted your question in the Verizon Cloud space, but then mentioned the Apple iCloud service?
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
What does it mean when it says I need a data connection when I try to delete a picture from iCloud
Just what it says - the cloud is internet based, so in order to access "the cloud" (whether Verizon, iCloud, etc) you need an internet (data) connection. It can be wifi, or cellular, but you have to have an internet connection to access cloud stuff.