The wallet is the place where your coins are stored. You can also send and receive coins. The wallet is also the place where you will find your walled unique identifier. This is like your account number: If someone likes to send you money, this is the number you need to provide to them.
A wallet is a tool running on your computer, mobile or tablet. There are also hardware wallets out there which mostly connect via USB interface to your computer.
Here is a list of some wallets you might try:
Wallet Platform(s) Homepage Comment Electrum Linux/OSX/Windows/Android https://electrum.org lightweight wallet Magnr Online https://magnr.com provides interest rate for your savings and trading platform Multibit Classic Java (Linux/OSX/Windows) https://multibit.org/releases.html In legacy mode Multibit HD Java (Linux/OSX/Windows) https://multibit.org Transaction fees involved
Articles comparing wallets:
Electrum vs Multibit: a bitcoin thin client comparison