What's the difference between off-network and on-network?

I'm try to work out which is best for what I want out of the service. I want protection while on my iPhone 5 (both when connected via Wi-Fi and when roaming on 3/4G).

Website with product details:
http://www.umbrella.com/products/umbrella-security-features/

Off network means you are trying to connect to a network other than your service provider. On network means your are communicating with a device in your service provider network. For security have AVG installed in your I phone against threats. Hope this answers your question.