Cydia is a software installation app available for iOS devices (such as the iPhone or the iPad) which enables the user to download and run software not available through Apple. Cydia is the creation of developer Jay Freeman, also known as 'saurik.'
Cydia can only be installed on devices that have been 'jailbroken.' To 'jailbreak' an iOS device means that a program has been used to remove the limits on your iOS device which have been installed by Apple.
This then allows the user to install software which would otherwise have been blocked by the device's security settings. Cydia can also install its own interface (seen in the picture above) with which the user can interact.
Apps offered in Cydia may have been rejected by Apple as they violate the company's terms or directly compete with Apple’s own apps.
What Can I Install Using Cydia That I Can't Get from Apple?
There are a great many apps and 'tweaks' (minor programming improvements) only available through Cydia that users believe to be essential; games, productivity and networking applications, even the look of your iOS device - the icons and wallpaper, for example - can be changed once Cydia has been run.
Once installed, Cydia presents the user with a 'store,' much like the Apple iStore, listing new applications for you to buy (Cydia itself is free to download).
Here's a YouTube user's guide to Cydia: