An iPhone can be backed up without a computer by wirelessly connecting to iCloud, and once an initial connection is made, iCloud will automatically backup any connected device at least once a day provided certain conditions are met. Using iCloud may require upgrading if the initial storage capacity of 5GB runs out, however.
In order to allow iCloud to successfully backup an iPhone, the device must be locked, connected to a power source, and still turned on. Wi-Fi also needs to be available. Providing these conditions are met, then iCloud will backup the device every day. Almost everything on the phone will be backed up, including settings, ringtones, apps, and any photos or videos present on the device.
Given that some information is already stored on iCloud (contacts, address book, etc), it is possible to quickly run out of space if usage is not monitored. This is especially true if there are a lot of large files (photos, videos, and music for example) on the phone that get updated or added to frequently. It is also possible to reduce the amount of data that iCLoud chooses to store, so selecting only the most important files and data to store is possible in order to save on space in the long run.