This is the central mechanic behind the bitcoin economy, and mining is used to keep transactions secure and reliable. Bitcoins are stored in digital wallets that are encrypted to protect your money. These wallets can be stored either locally or online. Online services are generally considered less secure as your money could potentially be lost if something catastrophic happens on their end. Local wallets typically require verifying the entire blockchain, which is the history of all bitcoin transactions. Hosting a blockchain is what helps keep Bitcoin running and secure.
Syncing this blockchain for the first time can take a day or more. Popular local wallets include BitcoinQT, Armory, and Multibit. Multibit does not require downloading the entire blockchain. You can also get wallet apps for your mobile device. These do not require downloading the entire blockchain.