In the Ethereum universe, there is a single, canonical computer (called the Ethereum Virtual Machine, or EVM) whose state everyone on the Ethereum network agrees on. Everyone who participates in the Ethereum network (every Ethereum node) keeps a copy of the state of this computer. Additionally, any participant can broadcast a request for this computer to perform arbitrary computation. Whenever such a request is broadcast, other participants on the network verify, validate, and carry out (“execute”) the computation. This causes a state change in the EVM, which is committed and propagated throughout the entire network.
Requests for computation are called transaction requests; the record of all transactions as well as the EVM’s present state is stored in the blockchain, which in turn is stored and agreed upon by all nodes.
Cryptographic mechanisms ensure that once transactions are verified as valid and added to the blockchain, they can’t be tampered with later; the same mechanisms also ensure that all transactions are signed and executed with appropriate “permissions” (no one should be able to send digital assets from Alice’s account, except for Alice herself).