A Bitcoin wallet is a software program that stores bitcoins. Technically it does not really store the coins but allows you to access your coins on the public ledger (called the blockchain) by storing your private keys (which ou MUST keep secret)

Bitcoin wallets also allow you to send and receive bitcoins.

The difference between a hot wallet and a cold wallet is that hot wallets are connected to the internet, while cold wallets are not. Hot wallets are more vulnerable to online attacks but because they are online they make it easier to buy/sell your bitcoin. Cold wallets are more secure but less convenient.

Most web based wallets, mobile wallets and desktop wallets are hot wallets. We woiuld strongly advise you not to store large amounts of money in hot wallets.

Hardware wallets and paper wallets are both examples of cold wallets

Hardware wallets use a physical medium (e.g. they often look like a usb stick) to store your private keys making it much harder for hackers to steal your bitcoin. i.e. they are not connected to the internet.

The vast majority of hot wallets are free - however there is usually a price tag of around 100 $/£/€ for cold wallets and of course this could feel very expsenive if you are storing a tiny amount of bitcoin and cheap if you have a large number of bitcoin stored in your wallet.

It is sometimes easier to think of a hot wallet like a pocket wallet for small change and a cold wallet as bank vault for long term secure storage.