A wallet is a collection of accounts, and an account is a combination of an address and its associated private key.
A wallet is said to be an online wallet when it is connected to the Internet. For example, wallets stored in geth, any website/database, and so on are called online wallets. Online wallets are also called hot wallets, web wallets, hosted wallets, and so on. Online wallets are not recommended at least when storing large amounts of ether or storing ether for a long time because they are risky. Also, depending on where the wallet is stored, it may require trusting a third party.
For example, most of the popular wallet services store the private keys of the wallets with themselves and allow you to access the wallet via an e-mail and password, so basically, you don't have actual access to the wallet, and if they want, they can steal the funds in the wallets...
