At a high level, a wallet is a software application that serves as the primary user interface to Ethereum. The wallet controls access to a user's money, managing keys and addresses, tracking the balance, and creating and signing transactions. In addition, some Ethereum wallets can also interact with contracts, such as ERC20 tokens.
More narrowly, from a programmer's perspective, the word wallet refers to the system used to store and manage a user's keys. Every wallet has a key-management component. For some wallets, that's all there is. Other wallets are part of a much broader category, that of browsers, which are interfaces to Ethereum-based decentralized applications, or DApps.
One key consideration in designing wallets is balancing convenience and privacy. The most convenient Ethereum wallet is one with a single private key and address that you reuse for everything. Unfortunately, such a solution is a privacy nightmare, as anyone can easily track and correlate all your transactions. Using a new key for every transaction is best for privacy, but becomes very difficult to manage.
A common misconception about Ethereum is that Ethereum wallets contain ether or tokens. In fact, the wallet holds only keys. The ether or other tokens are recorded on the Ethereum blockchain. Users control the tokens on the network by signing transactions with the keys in their wallets.
<aside> 💡 Ethereum wallets contain keys, not ether or tokens. Wallets are like keychains containing pairs of private and public keys. Users sign transactions with the private keys, thereby proving they own the ether. The ether is stored on the blockchain.
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There are two primary types of wallets, distinguished by whether the keys they contain are related to each other or not.
The first type is a nondeterministic wallet, where each key is independently generated from a different random number. The keys are not related to each other. This type of wallet is also known as a JBOK wallet, from the phrase "Just a Bunch of Keys".
The second type of wallet is a deterministic wallet, where all the keys are derived from a single master key, known as the seed. All the keys in this type of wallet are related to each other and can be generated again if one has the original seed. There are a number of different key derivation methods used in deterministic wallets.
To make deterministic wallets slightly more secure against data-loss accidents, the seeds are often encoded as a list of words for you to write down and use in the event of an accident. These are known as the wallet's mnemonic code words. Of course, if someone gets hold of your mnemonic code words, then they can also recreate your wallet and thus gain access to your ether and smart contracts.