When you store bitcoins, you store private keys that let you access your bitcoin address. Bitcoin wallets let you store these private keys just as you would store cash in a physical wallet. There are three main ways to store bitcoins: on your PC, on your mobile device and on the web.
How to store bitcoins on your PC
In order to be able to store bitcoins on your computer you need to download one of the bitcoin wallets (software). Generally, it is much safer to store the currency on your own computer as you are the only one in control of the software. The downsides of storing bitcoin on your PC would be 1) Deleting the wallets on accident/computer breaking of accidental disposal 2) Your bitcoin wallet stays at home so you will not be able to use it if you decide to buy something in a shop. There are different types of bitcoin wallets and the one you choose depends on your specific needs.
Bitcoin-Qt (download via sourceforge) is a full bitcoin client which comes with its own bitcoin wallet. It is very secure, private and stable but has less features and takes up more space and memory.
MultiBit (https://multibit.org/) is a more lightweight version of the bitcoin client. It synchronizes with the network very quickly and can be ready for use within minutes of the installation.
Armory (https://bitcoinarmory.com/) is an advanced bitcoin wallet with many extra features. It allows you to backup and encrypt your private keys as well as store them on an offline computer.
The wallets above are the most popular ones and there are others available. As time passes by more advances bitcoin wallets would appear.
Mobile wallets allow you to bring your bitcoins with you. With mobile wallets you can easily exchange the currency if you are meeting with someone in person or paying for an item in a physical store. You can send bitcoins using a mobile device by scanning a QR code, via NFC or by typing in the actual bitcoin address. The most popular mobile wallets are:
Bitcoin Wallet (play store) is a decentralized bitcoin wallet for android and blackberry. It is very simple, easy to use, reliable and secure. Allows you to send bitcoins offline through Bluetooth.
Mycelium (play store) is another mobile wallet. It allows you to secure your wallet with a pin, encrypt your keys on a PDF document and restore them if needed. Private keys can be imported for cold-storage.
Coinbase (play store) is another easy to use web wallet that also provides its users with an android app. You can exchange you USD to bitcoins and vice versa through Coinbase.
Web wallets tend to be less secure than some software and mobile wallets since they are more prone to hacking. If a web wallet service gets hacked the attackers might get access to your private keys and bitcoin funds. It is always recommended to use the extra security features that the web wallets offer in order to protect your bitcoin funds. The most popular web wallets are:
Blockchain.info allows you to store an encrypted version of your wallet online, has two factor authentication, cloud backups and payment notifications. Blockchain also offers a mobile app which can be convenient if you intend to use a mobile wallet as well.
Coinbase – An easy to use wallet services. Allows you to link your US back account to deposit/withdraw your currency.