A Hot Wallet is a wallet that is connected to the internet or where the private key is stored on the machine that is plugged into a computer and connected to the internet. Most people keep a small amount of money in their Hot Wallets for purchasing tokens or doing transactions (similar to real wallets that people use on a day-to-day basis). You personally sign each transaction.
Cold Wallets are used for long-term storage of cryptocurrencies and typically hold large amounts of money that are not accessed very frequently. Cold wallets are only ever connected to the internet to make a transaction and the private key never leaves the physical device, which decreases the likelihood of these funds being stolen by a hacker. You personally sign each transaction. Trezor and Ledger are commonly used cold wallets.
Any funds held in exchanges like Bittrex and Poloniex are stored in these companies’ servers and are connected to the internet to be actively traded. Storing funds in exchanges can offer added risks, as you do not hold the keys to your wallet, a third-party does. Make sure you review and trust any exchange before storing funds there.