First, don't confuse the terms "offline" vs. "online", use "bitcoin wallet" and "bitcoin account" instead. All wallets that are not considered "cold storage" are "online" because they are connected to the internet.
Bitcoins reside in the block chain.
A bitcoin wallet contains the keys that allow you to spend the bitcoins.
A bitcoin account is an account with some service (such as an exchange) that holds bitcoins for you. They hold the keys to the bitcoins in their wallet and your account keeps track of how many of the bitcoins in their wallet are supposed to be yours.