A hot wallet refers to a Bitcoin wallet that is online and connected in some way to the Internet. It is a term that refers to bitcoins that are not being kept in cold storage.
Bitcoin-related services and exchanges that are able to pay out withdrawals instantly can be said to be paying them from a “hot wallet”. The term can also be used loosely to refer to keeping bitcoins in an exchange where they can be withdrawn on demand. Its real-world analogy is keeping cash on person: easy access, but greatest risk of unrecoverable theft in the event of an attack.
Operating a “hot wallet” is also a risk to its owner, because most computer systems have hidden vulnerabilities of some sort that can eventually be used by hackers or malware to break into the system and steal the bitcoins.
Keeping large amounts of bitcoins in a hot wallet is a fundamentally poor security practice. Most, if not all of the Bitcoin losses incurred in all the known hackings in Bitcoin history can be attributed to funds kept in hot wallets.
Most reputable services offering Bitcoin withdrawal of some sort will keep a very limited number of bitcoins in a hot wallet to enable immediate withdrawals of small/typical amounts, but will require a delay and manual activity to process a larger withdrawal, as the bitcoins are retrieved from other storage.