Peer-To-Peer Digital Currency
First of all, whatever bitcoin software you decide to download, don't forget to backup your wallet! (For more tips on keeping your bitcoins safe, see bitcoinsafety.com)
The best place to download the original bitcoin client software is from bitcoin.org. However this client software is not necessarily the best thing for people who are new to Bitcoin as it can take a long time to set up and can take up many of your computer's resources. At bitcoin.org you will find a few different alternatives for using bitcoin directly.
Other bitcoin clients that may be worth downloading are: Electrum, Multi-bit, Armory just to name a few. These are all linked to from bitcoin.org.