A Bitcoin wallet is a file that stores a collection of private keys that you need to access your bitcoin/ altcoin address and spend your funds. These private keys are the value of all your Bitcoin or Altcoin in your account. A Wallet is a free service which lets you send and receive Bitcoins or Altcoins.
The best Wallets are secure and protected as your private keys are encrypted with your password. The Wallet Company will not know your password, and therefore are unable to view or spend your Bitcoins/ Altcoins. You retain full control of your private keys, so your wallets can never be seized or blocked and can be imported into any desktop Bitcoin client you choose.
They come in several different forms, designed for different types of devices.
Advanced miners can use alternative paper storage to avoid having their bitcoins/altcoins on a computer at all. Of course, it is very important to secure and back up your bitcoin wallet.
Bitcoins are a modern equivalent of cash and every day, more and more new merchants start accepting them as payment.