What are crypto wallets?
In crypto, people talk a lot about wallets and keys. Some of the particular types, hard wallets, soft wallets, cloud wallets, multi sigs and even brands, like MetaMask and Phantom, get a lot of prominence.
So, to start with crypto- at its core, is a registry where an entry is changed by someone holding the private key, applying the proper digital signature to the registry. To do this, requires software which can access nodes/miners, and then manipulate the change of a registry entry with a valid signature. The wallet may also do other things, like show all assets held by its owner and have interfaces to exchanges to allow transactions.
Within this world, there are lots of options. Typically, a wallet's private key is a 1024 or 2048 bit string that pairs with a known public key (same length). To make it easier to remember, most wallets use seed phrases, which select from a sampling of 1024 words and their arrangement maps to a 1024 bit private key (longer keys have more words in the seed phrase).
This private key is either entered or generated by the wallet. And, it's transportable. This is good as it lets you easily move between wallets -- but don't lose the phrase, or else someone can easily take all of your assets.
In addition, wallets may have different security measures built in. A hard wallet is a piece of hardware that allows or is required for manipulation. This is beneficial because it means that any hack or theft requires physical theft, not just online hacking. And, given that most attack vectors to crypto are through the internet, it makes it much harder for someone to gain access to your account. In addition, some wallets are designed so multiple users need to sign off. This is helpful to stop internal theft, such as a rogue employee stealing assets, and means that someone stealing seed phrases or passwords would need to steal two people's, not just one.
As Web3 evolves, and DeFi grows, many applications now require wallets. This is because of the Web3 ethos that you own and can take your identity with you. What this means in practice is that you can connect and disconnect your wallet when you feel like it, and the applications themselves are otherwise stateless, until a user shows up with a wallet. If you use DeFi applications, this will become immediately common, when MetaMask pops up (on Eth) or Phantom (on Solano) or other wallets.
However, some web applications are wallets themselves, where the keys are held in the cloud (such as Blockchain.com) or someone else manages the keys and never shows them to you (as in Coinbase).
This makes the wallet an incredibly important part of the DeFi experience. It provides a huge amount of security and functionality. The many variations exist because of the different L1s, but also because of the use case, trading off security for convenience and allowing for various different types of user interfaces. In fact, if you used some of the earliest wallets for Bitcoin, you can already see how far the user interfaces have come. However, when you compare the experience of a wallet to many native web applications, there is still very far to go and we expect to see major innovation here in the future. And, we'll all be better off for it.