A cryptocurrency is decentralized replicated blockchain-based ledger system. What is ledger? A ledger is a state of a system in some minimal form giving us ability to answer whether a transaction could be valid against it or not. Another question a ledger should answer is whether a transaction is already included in it or not. How could be ledger represented then? There are two popular approaches.
First is used in Bitcoin and its successors. A ledger in Bitcoin could be represented as unspent transaction outputs(UTXOs in Bitcoin jargon) list. Given a transaction it is easy to check whether it is valid: all the transaction inputs must be connected to unspent outputs, and also inputs must provide a valid solution(in form of stack machine script) to spending guard condition an UTXO has(in form of stack machine script as well). Also, sum of bitcoins associated with inputs must be not less than sum of bitcoins associated with inputs. Outputs spent are to be removed from ledger and new ones the transaction contains are to be added to it.
Abstracting the Bitcoin-like model, a state could be represented as a list of closed boxes. Each box has a value associated with it. A transaction contains keys to open some boxes and new closed boxes. A transaction is valid if all the keys in it are opening closed boxes in a ledger and sum of new closed box values is no more than sum of values of closed boxes to be open.
It is not hard to see that the second question (whether a transaction was already processed into ledger or not) could be answered trivially in the box model. If transaction was already processed, its boxes are open and so not stored in ledger anymore.
Unfortunately, there are no known box systems other that Bitcoin-like around, but I bet, we’ll see them around some day.
While boxes state model is about immutable objects to be only created or destroyed, we can also represent state with mutable accounts. Transaction is involving existing accounts. It modifies them and maybe creates some new accounts. This approach is adopted by Nxt and Ethereum. In a simplest form, it is implemented in Simplest Transactional Module of the Scorex project. Basically the state in Simplest Transactional Module is just map (account -> balance).
While this model could answer pretty well to the question whether a transaction is valid against a state(in the simplest case above, it is just about whether an account has an aprropriate balance), the second question(whether a transaction was already included in included into state or not) becomes tricky. As a possible solution, nonce could be used, and is used in Ethereum. That is, a transaction contains always-increasing nonce(so a next transaction must have a bigger value of nonce field than a current one), and last nonce value used is to be stored into state.