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From my understanding there are applications running "on the Ethereum blockchain" and these applications are available through Mist.
What does it mean that the applications run on the Ethereum blockchain? Where is the data for these applications stored? I understand that some data such as account balances for these applications may be stored on the blockchain, but what about the rest of the application?
You can use whatever storage mechanism you want depending on your requirements.
One of the popular choices is IPFS because the way how content is identified in IPFS (by using its cryptographic hash) naturally fits into Ethereum smart contracts. So you keep the hash of a file in Ethereum smart contract and the content itself in IPFS.
The application (Dapp) is a front end plus a smart contract. The bytecode of this later is stored in a contract account in the blockchain. The front end could be a simple javascript module or a webpage using web3js framework. This front end is not stored in the ethereum blockchain it could(not necessarily) be stored in a decentrlized storage network as ipfs or swarm. As you might know to brows such webpagea wich nees to communicate with an ethereum node(send transactions,..) you need to use a special browser like Mist or a plugin in your default browser like Metamask.
