Once you purchase your Bitcoins you need to decide how and where to store your asset. One option is to keep it on the exchange where you purchased the Bitcoins. This has its pros and cons which will be discussed in this article. Other options include paper wallets, hardware wallets. The issues surrounding these offline storage will be discussed in another article entitled, "Alternative options for storing your Bitcoins and other cryptocurrencies"
Storing your Bitcoins on an exchange:
One of the advantages of keeping your cryptocurrency on an exchange is ease of access. Sites like Coinbase have an easy to use interface and can be linked directly to your bank accounts. This makes both the purchase and sale of cryptocurrency easy and transfers between your cryptocurrency account and bank relatively seamless. They also offer access to software that allows you to determine your profits / loss for tax calculation purposes.
Leaving your Bitcoin on an exchange requires a degree of trust. There have been numerous sites that have been hacked. In addition, some sites appear to have committed outright fraud stealing their customers cryptocurrency. Unlike bank accounts which are FDIC insured, cryptocurrency exchanges are poorly regulated and your crypto assets are not insured by any government agency. So if your exchange closes down, goes bankrupt or just steals your money, your only recourse is to seek a legal remedy. In most of these cases the customer money is lost. However, there is increasing regulation and sites like Coinbase.com are well respected and your assets are likely safe there.
Cryptocurrency Hardware Wallets
Hardware wallets allow you to store the code for your Bitcoins (the private keys) on a secure electronic device. These devices are designed to only allow you to relatively safely store your Bitcoins offline and away from hackers. Usually they require you to have a secret phase to unlock the device and release the coins. The link below has some examples of hardware wallets.
Paper Wallets to store your Bitcoins is also an option. This topic is covered in a separate article on this site.