The concept of wallet types is a function of security vs convenience. Some wallet types are less convenient, but more secure. Others are more convenient, but less secure. Choosing between wallet types depends on the use case and value stored in them.
In general terms, smaller amounts of crypto and high frequency use are better managed in less secure wallets such as software hot wallets. This is because the crypto is always ready to use and sending transactions is as simple as a touch of a button from a computer or phone.
Larger amounts of crypto and low frequency use are better managed in highly secure wallets such as hardware and cold wallets.
Wallet types are defined by where the private keys are and in what format. All private keys are used to control addresses inside blockchains, and these addresses are the ones that hold people's cryptocurrencies and crypto assets. This is why the location of the private keys is what constitutes the type of wallet.
What is a hardware wallet?
A hardware wallet holds the private keys in a separate hardware device that never touches the internet nor your computer or phone.
The private keys are kept in a sort of pen drive that is connected to your computer or phone with a cable only when a transaction needs to be signed. However, the transaction is signed inside the hardware wallet, so the private keys never touch your machines or the internet, where hackers may gain access and steal your crypto.
Because of this feature, hardware wallets are considered more secure than software wallets.
IMPORTANT: A HARDWARE WALLET IS NOT A COLD WALLET PER SE. A HARDWARE WALLET MAY CONTAIN COLD WALLETS OR HOT WALLETS, THOUGH. IT ALL DEPENDS ON WHETHER THE PRIVATE KEYS HAVE BEEN USED TO SIGN TRANSACTIONS. PLEASE SEE THE DEFINITION OF COLD AND HOT WALLETS BELOW.
What is a software wallet?
Software wallets hold the private keys encrypted inside your computer or phone. Usually, they are encrypted with a password you select to use the wallet app you are using, such as Emerald.
Although this high level of encryption is extremely secure, machines and phones may be lost or hacked and there is a chance that someone else may gain access to your crypto if they can guess or reverse engineer your password.
For these reasons, software wallets are considered less secure than hardware wallets. However, they are more convenient to use because they don't require connecting your hardware or cold wallets each time you need to transact on the blockchain.
What is a cold wallet?
A "cold wallet" is the highest level of security in terms of managing private keys. It holds private keys associated to addresses on the blockchain where the private key is on paper or on a hardware wallet, but has never been used to sign a transaction.
This is considered the highest level of security in the blockchain industry because once you sign and send a transaction to a blockchain there is some traceability and information now on the public domain that could conceivably, however unlikely, be used to access your crypto assets.
Although improbable, this marginal risk is sufficient to warrant keeping large amounts of crypto in this type of wallet, however inconvenient they may be to manage.
When a cold wallet is used to sign a transaction, then it becomes a "hot wallet", even if it's on a hardware wallet or a paper wallet.
If a private key is on a software wallet, it is "hot" even if it has never been used to sign a transaction as it is connected to the internet inside your machine or phone, which a less secure environment.
What is a hot wallet?
A hot wallet is one in which the private keys associated to addresses on the blockchain have been used to sign one or more transactions regardless of whether it is in a hardware wallet or a software wallet.
Any keys, used or not, on software wallets are considered hot wallets because they are readily available to sign transactions and are in a less secure environment than hardware wallets or cold wallets.