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.
Multibit is a Java-based wallet for your Bitcoins for Linux, OSX or Windows
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