I have made this simple for quick reads so here it goes...
We have read news that exchanges keep getting hacked in the news, so if you are leaving your cryptos in there there could be some risks.
Best practise is that you keep it there when you are trading. If you are not trading, put them into a separate wallet like a cold wallet for keeping it long term. Explanation as follows:
By abstract:
Hot wallets, frequently used.
Cold wallets: (e.g.hardware wallets)
* Wallets that are kept strictly offline, kept long term.
* In cold wallets, public and private keys are generated offline
Brain wallets definition: "A brain wallet is a standard wallet that generates its address by hashing a passphrase to create a private key and therefore a public key and resultant address." - source: https://www.cryptocompare.com/wallets/guides/what-is-a-bitcoin-brain-wallet/
Public keys (Share. For the public, to be used to receive deposits, it can be a long list address or could be a qrcode)
Private keys (Secret. For signing which the owner does not reveal publicly)
Brain keys (this one slightly different from a brain wallet but of a similar nature, but most commonly you see the pattern of a number of blocked keywords to be used as a passphrase to sign, along with your username and password)
e.g. apple cat espanyol esquire calculator forge sword......
This one, looks something like a long list of blocked keywords that performs the same way as the private key/passphrase.
Getting More Defined:
Type 1: Desktop Wallet
- recommended: e.g. "Armory" (high security).
- others: e.g. "Multibit", "mSIGNA", "Hive".
Type 2: Mobile Wallet (with QR codes)
- "Mycelium", "Blockchain" for android / IOS.
- "Bitcoin Wallet" for Blackberry.
Type 3: Online web wallets: (Web based wallets, usually for starters)
- e.g. "Coinbase", "Circle", "poloniex"
- e.g. "Bitgo" (trusted by institutions, co-sign to lock accounts)
- (connected to the internet, private keys are generated by the computer connected to the internet)
- e.g. "Coinkite" (multi-signature wallets)
Type 4 Physical wallets
Paper wallet (a long string of words written on paper)
- generate via "bitaddress.org" or "blockchain.info", then print out on a piece of paper.
Type 5 Bitcoin Clients
- "BitcoinQt" (first ever wallet created, believed likely used by satoshi nakamoto)
- "Electrum" (lightweight client)
Type 6 Hardware Wallets
- e.g. Trezor / Ledger Nano S
- (Trezor is also working on integrating more altcoins, at this moment is supports BTC,ETH,ETC,DASH,ZEC and LTC)
ref source from : 
coinoutletatm.com
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