For many users it will be appropriate to run the Bitcoin core client or Bitcoin Qt. Provided by the core Bitcoin developer team, BitcoinQT provides a wallet and can also act as a full node, downloading and maintaining an up-to-date copy of the blockchain and thus, contributing towards overall network stability.
Some users will opt to utilise an online wallet solution; there are a number of great options across the web. Such wallets are highly flexible and often accessible anywhere you can get online; these kinds of wallet are often combined with other services such as Bitcoin exchange capabilities. There are two distinct categories of online wallet, one type gives users access to private key information, such that they have full control over their coins. The other type hides a way this information, keeping life easier for the user, however the drawback of this approach is that the user is wholly reliant on the security systems the wallet provider has in place. Luckily, most companies which provide such services are large, well established Bitcoin businesses with solid security practices.
Mobile app based Bitcoin wallets are also now available for both iOS and Android from multiple different providers. It is worth noting that many of the different Bitcoin storage methods are focussed towards a different user demographic, for example those who are particularly security conscience or more active Bitcoin users.
Bitcoins which are not going to be used in transactions for some time are best off in cold storage, offline. There are a few different ways to ensure bitcoins are securely put into cold storage, many users opt to utilise a separate laptop which stays offline. This is the safest, most secure way to store bitcoins, but its security is only ensured if the user keeps the device offline and has a consistent methodology when using it. Most large online wallet providers who hold Bitcoin for the user will keep a portion of their overall stores in cold, offline storage to ensure its safety.
Hardware wallets are also an option for arguably the best security possible; a few systems exist on the market, normally around the size of a USB memory stick, these devices sole function is to act as a secure, portable Bitcoin wallet.
There are a huge range of viable options which will suit a number of different types of user; often the best way to find the best wallet is to try a few different options out to see what works best for you. A number of options available are introduced on the Bitcoin website, but these represent only a few of the huge number of possibilities.