A wallet generates an ether address that you can use to receive and store ether. You can store as many ether (ETH) as you want in as many addresses as you want.
Public Key Cryptography
Ethereum wallets are based on public-key cryptography, where the Ethereum protocol creates a private key and public key with the Elliptic Curve Digital Signature Algorithm (ECDSA). The wallets private key is used to sign a transaction and this signature mathematically proves that it is genuine, was signed by the owner and prevents the transaction from being altered.
Private keys can generate lots of public keys, where public keys are put through a hash function to generate public addresses. These public addresses can be used to receive ETH. To spend ETH, you need to control the private keys, hence why you should keep them a secret.
Once a transaction is signed with the private key, it is added to the mempool for miners to confirm it. Miners will use the sender’s public key to guarantee the signature is authentic and include the transaction in the next block if it checks out. Once the transaction is included in the next block, the ETH is transferred from one wallet to another.
Different Types of Ethereum Wallets
Ethereum wallets differ to Bitcoin's as the smart contract functionality means that wallets can interact with smart contracts by either creating them or triggering them.
A further difference is that Ethereum wallets can hold cryptocurrencies other than ether, specifically ERC-20 tokens that are based on the Ethereum blockchain (but these tokens will have a different address).
There are many different types of ether wallets, and they have different trade-offs regarding convenience and security.
The official ether wallet includes the Ethereum clients, which can be downloaded here.
These clients allows you to run a full node so you can verify your transactions and support the network by validating transactions and blocks. Ethereum clients are the most secure desktop wallet available, but there are other desktop wallets that can store ether, such as MetaMask, Exodus, and Electrum.
You can also store ether online using services like Blockchain, but these are not as secure as desktop wallets and have suffered security breaches in the past.
If want to use online ether wallets, then it is best to store only small amounts and enable 2FA.
Mobile wallets are convenient when using and spending ether. Your smartphone’s camera can read QR codes, which represent ETH addresses, to spend ether.
Offline wallets, which include hardware wallets and paper wallets, offer enhanced security as compared to mobile and online wallets. Offline wallets are also known as ‘cold storage’.
Paper wallets can also be generated by using paper wallet generators such as MyEtherWallet.
A paper wallet usually displays your public and private keys, and the challenge is then to keep your private key safe. Setting up a paper wallet can be complicated if you want maximum security, as you will need two laptops, one of which has never been connected to the internet.
Whichever wallet you choose, ensure you back it up so you can restore it if your computer is hacked or you cannot access your mobile wallet.
A backup is usually a secret combination of 24 words, known as a seed, or an encrypted version of your private key. You may also want to encrypt the seed backups, especially if you are storing them on your computer. Storing your private keys in several locations is the most secure option. You can store your private keys on paper, USB drives, and CDs.
To gain a deeper understanding of storing and securing your ether wallet, check out this lecture by Pamela Morgan.