Can I get short a explanation about cryptocurrency cold/hot wallets?

Hot wallets

These are any kind of wallet that has a network connection and so is vulnerable to attack and is relatively unsafe. Many people prefer to keep their savings in a cold-wallet and only transfer money to a hot-wallet when they need to spend the money.

You need a hot-wallet to send a payment
You don't need a hot-wallet to receive payments.

synonyms: online-wallet (confusingly - don't use this term)

Cold wallets

These are wallets that are not connected to the Internet, or any local network, and so are relatively safe from network-based attack by thieves.

A cold-wallet can receive payments.

Synonyms: offline-wallet.

Varieties: paper-wallet, hardware-wallet (at least when not plugged in). Desktop wallet on isolated or turned-off PC, ...

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Are there more kinds?

There are lots of ways of classifying wallets. Here's the main other kinds I know of:

Desktop wallet

A wallet that runs on a normal PC.

Mobile Wallet

A wallet that runs on a mobile phone. Almost always these are lightweight wallets.

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Custodial wallet

A wallet where the private-keys are held by an online service, not by the customer.

Typically access to the wallet is by a web-browser but sometimes with an app or using a custom desktop-client.

Advantages: easy and fast to set up, transfers to other customers of the same service can be instant (no wait for confirmations) and cheaper.

This is the least safe way to hold crytocurrency (see large scale thefts from MtGox, Poloniex, Bitstamp, BitFinex, Cryptopia, Quadriga-CX etc etc etc)

Key point to remember: "No private-key = no Bitcoins" or "not your private-key means not your Bitcoins".

Synonyms: web-wallet, hosted-wallet, online-wallet (rarely and confusingly)

Paper wallet

Wallets don't contain crytocurrency coins, the only important thing they contain are private-keys - anything else in them is disposable.

Therefore printing your seed-phrase or private key onto paper as a QR-code, other barcode or as a hex or base64 string etc, can act as a wallet.

A paper wallet can receive payments but cannot make them.

A properly-managed paper-wallet is one of the most secure types of wallet.

Hardware wallet

A hardware device that safely holds your private key and which can connect to a PC, usually using a USB port, to sign transactions (without divulging the private key to the PC)

This is arguably the most secure form of wallet.

They typically need software on the PC which knows how to talk to them. Several desktop-wallets can interface with one or other hardware wallets.

Brain wallet

NOTE: Brain-wallets can be very insecure. For more info see this question, or this one.

You don't hear this very often but it is an interesting form of wallet. There are standards for generating private-keys from a seed-phrase. Many wallets support this. If you can remember a seed phrase you can destroy your wallet and later re-create it using only the seed-phrase.

If your wallet is only stored as a memorised phrase in your brain, it can still receive payments - so long as you have your payment address(es) noted down somewhere. Payment addresses need not be kept secret.

Unfortunately people are pretty bad at remembering stuff.

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Full-node Wallet

A wallet that keeps a copy of the whole blockchain. E.g. bitcoin-core.

The trade-off is that the wallet takes days or weeks to synchronise when first installed. It will use up more of your network bandwidth (at least initially)

This is the type of wallet that was first invented and is in many ways, one of the best types of wallet.

Lightweight wallet

A wallet that does not keep a copy of the whole blockchain and which trusts some other nodes to give it truthful blocks.

The advantage is they don't need lots of storage space and can start-up much faster.

Synonym: SPV wallet.

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Watch-only wallet

This is not really a type of wallet but a way of using a wallet.

A watch-only wallet is one that has had a public-key / address imported into it but which does not have the corresponding private-key.

It can show the "balance" for an address but cannot be used to spend that balance.

Often used by scammers to trick people but also useful as a safe type of hot-wallet to monitor your balance without any risk of theft.

A wallet can usually have several sets of, erm, subwallets. A wallet could have both a private key for a hot-wallet with a small amount of money in it and an imported watch-only address for a cold-wallet so you can monitor balances for both "hot" and "cold" subwallets in one place.

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Things to remember ...

(... about wallets and their use)

• Wallets don't contain money.
  • The only important thing in a wallet is the private-key(s).
• Money is not sent to wallets.
  • Money is sent to bitcoin-addresses. The wallet contains a private-key that allows you to "spend" money at a corresponding bitcoin-address
• Never share a wallet-password - even if you change it afterwards, you've lost control of any money that has been, or might later be sent to addresses in that wallet.
• Never share your private-key. Never.
• Backup your private key or seed-phrase safely in several places.
• Share overlapping parts of your private-key or seed phrase with several family members with instructions on what to do if you are hit by a bus this afternoon. (see several questions here from desperate bereaved relatives of people who made no provision for this)
• What do you want to happen if your house burns down to ash with all your pcs, laptops, notebooks, phones, paper etc destroyed. (see questions here from people who lost their phone or overwrote their PC or mistyped their seed-phrase or forgot a password and don't know what a backup is or why they should have made several, in several places, in fireproof boxes, and tested recovery)

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Other lists

Other places that have lists of different types of wallet:

• Choose your wallet, Bitcoin.org
• Comprehensive Cryptocurrency Wallet Guide, "fastInvest"@Medium.com
• Types of Crypto Wllets, CoinSutra
• Is there a Wiki comparing different wallets?

Hot and cold in this case refer to if the private key is stored on a system connected to the internet or not. Hot wallets are stored on system connected directly to the internet. This can be on the client machine or on a centralized server. A cold wallet can be a HW wallet, an offline wallet or a paper wallet. You kind of create an air-gap that drastically reduce the attack surface on the wallet.