Cryptocurrency, for all practical purposes, started with Bitcoin. Bitcoin uses blockchain technology to assure a degree of anonymity. Due to the nature of the open blockchain, the transactions are visible to anyone with access to the blockchain. This allows the transaction to be both traceable and linkable. If the actual identity can be linked to an address, anonymity disappears.
The degree of traceability depends on the strength of the cryptography. True privacy is accomplished with advanced cryptography. Most cryptos do not offer any privacy beyond anonymity or pseudonymity.
Popular cryptos that offer basic anonymity would include Bitcoin, Litecoin, Ethereum, and Ripple, plus a host of others.
Private cryptos are designed to hide details of any transaction. They still have open public ledgers, while details of the transaction are hidden, thus protecting the identity of the owner. Everyone has the right to privacy which doesn’t mean that you have something to hide.
Popular private cryptos include Monero, Particl, Dash, PIVX, ZCash, and Verge XVG. These currencies use means such as Ring Confidential Transactions (RingCT), RingSignatures, coin mixing, and zk- SNARKS to ensure virtually unassailable privacy. The choice is yours. Anonymity, which protects you from most observation, or totally private transactions, which can be seen by no one.