State
The current record of who owns what and what every contract stores, produced by applying all past transactions in order.
The chain stores transactions; the state is the result of replaying them, and every block header commits to it through a state root so two honest nodes can confirm they computed the same thing. Ethereum keeps state in a Merkle-Patricia trie of accounts, balances, contract code, and contract storage, while Bitcoin's equivalent is the set of unspent outputs. State growth, sometimes called state bloat, is the problem that this set only ever grows as new accounts and contracts appear, steadily raising the disk and memory needed to run a node and therefore pressing on how many people can validate independently. Proposed responses include pruning historical data, charging rent for storage, expiring untouched state, and stateless clients that carry proofs instead of the full set.
In practice
An Ethereum block header includes a state root, so a node that processed the same blocks correctly will compute exactly the same root as every other honest node.
The common misunderstanding
State growth is not the same as chain size; a node can discard old blocks, but it cannot discard current state and still validate new transactions.