BITCOIN EXCHANGE is place where Bitcoin can be purchased. Basically if the Bitcoin is purchased or sold to obtain or give anything be it fiat money, tangible goods or other services its considered as a Bitcoin Exchange.
Bitcoin can be bought or sold using fiat cash, electronic transfer or any tangible element. Bitcoin exchanges operate just like a money exchange company, just that Bitcoin Exchanges are more cryptographic and transactions can be done easily almost any where in the world.
Bitcoin Exchanges provide you with a Wallet where the BTC which you purchase or sell is store. Please bear in mind that these are not you personal wallet but perform same function as a any typical BTC wallet. Its most recommended that your Bitcoins are stored in you own personal wallet.
Which exchange you use is mostly depends on where you reside and which place is more accessible to you, provided they are trust worthy. Well! Ya gotta do you home work.
To get started you can check out here or here to know more & register and feel experience of using an exchange.