A wallet is an app or website that can be used to receive, store and send cryptocurrency only.
An exchange is an app or website that can receive cryptocurrency and fiat currency like USD, store it and convert it to another currency before sending.
Examples of wallets:
Blockchain.info : The most popular wallet and block-explorer site in the world, it has an android and IOS app that can be connected to the website to be used simultaneously
Blocktrail.com : Renamed recently to “BTC.com” https://btc.com/ after the most recent update they also have a wallet for android and IOS.
Mycelium app : Easy to use, offers the option to store bitcoinin USD or Euros.
Copay app Allows "multisignature" two or more separate passwords required to spend
Jaxx app : Accepts the most trusted currencies and bitcoin
Coinomi app : Native support for bitcoin and most major altcoins and tokens.
Examples of exchanges:
Kraken.com has an app can be used from most states in the USA
Virwox.com Accepts paypal Gemini.com Accepts cards, fastest payouts in USD, accepted in New York
Paxful.com Accepted anywhere for almost any type of payment even paypal, the price will me up to 30% higher
Bitwala.com every country not USA
Coinjar.com : An exchange available for Australian players
Shapeshift.io Has an app for Androis and IOS, No need to create an account in order to convert.
Libertyx.com : already has 19,000 retail locations in the USA that allow customers to buy Bitcoin with cash, and this number is steadily increasing.
Please keep in mind:
For long-term storage of bitcoins (or giving as gifts) it's not safe to store your bitcoins in an exchange or online wallet. These types of sites are regularly hacked. Even keeping a live wallet on your own computer or smartphone can be risky.
“Cold storage” refers to any means of keeping your private keys safe from hacking by holding them offline, some examples are:
Trezor Regarded as the safest and most popular hardware wallet
Ledger Most economic hardware wallet, still better than other types of wallets that are not hardware based.
Keepkey can store bitcoin, ethereum, litecoin, dogecoin, dash, and namecoin
A bitcoin paper wallets is just the Bitcoin private key printed on a piece of paper.
How to create paper wallets:
https://themerkle.com/top-4-ways-to-generate-a-bitcoin-paper-wallet/
https://www.weusecoins.com/bitcoin-cold-storage-guide/
https://www.bitcoin.com/guides/setting-up-your-own-cold-storage-bitcoin-wallet
https://en.bitcoin.it/wiki/Paper_wallet
https://www.coindesk.com/information/paper-wallet-tutorial/