A bitcoin wallet, is a piece of software that hold your private keys and through that controls your bitcoins.
A bitcoin wallet is typically an open source piece of software that has been peer-reviewed by the community as a hole.
There is a few different types of wallets, that let you hold your bitcoins. The first is a Web Wallet, where you let a webservice hold and control your bitcoins. Please be advised that you trust the security and integrity of the webservice, not to steal your bitcoins and in general to keep them safe.
So if you want to be in total control of your private keys and by extension, your bitcoins, we recommend that you use one of the following wallets:
• For Iphone: Breadwallet
• For Android phones: Mycelium
For laptops or desktop computers, you have a few viable options to choose from. First you will need to decide if you want to run a full node or a thin client, but if you want to get started in a fast and easy way, we recommend that you go ahead and use a thin client.
• Thin Client: Electrum
• Full node: Bitcoin Core, Bitcoin XT or Bitcoin Ultimate
To read more about the different full nodes, you can click here.