There are two methods for backing up your iOS install: one using iCloud, and one using local storage on a Mac or Windows machine using iTunes.
The iCloud backup is ideal for people who don't sync their device to a computer at all, or who have bought some additional space on iCloud (beyond the 5GB free space) if they have multiple iOS devices they want to back up. The iCloud sync preserves almost everything (pictures, videos, iCloud mail, apps, music and so forth), but some minor things like text-message logs or game high scores MIGHT not be included. The beauty of the iCloud backup is that it happens in the background routinely when your phone is connected to Wi-Fi.
The iTunes (local) backup backs up everything, since you have far more available space on a computer than you probably do in the cloud. This is best for people who sync regularly with iTunes and who need absolutely every scrap of everything on the iDevice backed up. It needs to be done manually by connecting the iOS device to iTunes, which can be as simple as opening the program on your Mac or PC while the device is connected by USB cable.