Cardano dApps

Hi everyone,

I think one of the most important aspects of blockchain will become the realization of dApps.
However, I do not quite get, how you would run them “on the blockchain”.
I mean, in case of Bitcoin, we have a big data base saved on the hard drive of each node.
You collect and verify transactions, put it into a block, calculate Merkle tree, hash the header (that includes the previous hash).
Then, you throw the dice to get a specific number starting with some zeros. If you succed you broadcast the block (with all the transactions) to the network.

Now, my question:
Where to “sneak in” the dApp?
Is the dApp executed on each single node? Then, the disk space of ETH or Cardano nodes disk space will get pretty problematic, I guess?
How to make sure, that the dApp code is executed? Would it not be possible to code a simple protocol to extend the chain without the additional run of power consuming dApps, if there is no “proof of dApp execution”?