Time Machine stuck at 'calculating changes'
One quirk with Time Machine that some people have experienced is when backup runs get stuck at "Calculating Changes" instead of completing the backup.
Time Machine has a great reputation as being a solid and robust backup system that can be used in a number of ways, however, as with all systems, it does have a few quirks. For example, sometimes backup runs can get stuck and not continue the backup. In these instances, though one or two backups may work fine, the system will start pausing at the "Calculating Changes" stage and never complete additional backups.
If this has just started happening, try just waiting it out. When there are major changes to the filesystem, Time Machine will do a "deep traversal" of the system to see which files have been changed, instead of relying on the fsevents cache to indicate what files need to be backed up. A deep traversal can take some time, especially on systems that have a large number of files, many of which can be within packages such as applications, installers, or application libraries.
Another approach may be to try repairing permissions. Though Time Machine should be able to back up all files on the disk, odd permissions on files and directories may interfere with this. Run Disk Utility when booted off the internal boot drive and run a permissions repair. Do not use Disk Utility from another boot volume for this task, as it will not load the custom permissions database for the current system, which may result in permissions not being fixed properly.
If waiting and permissions fixes do not work, try reapplying the latest Combo updater for OS X. Though rare, a faulty installation can result in odd problems with services like Time Machine, and a reinstall with the latest update is an easy and seamless way to fix this. Download the Combo updater for your version of OS X from Apple's Support Downloads page and re-run the installer when you have booted into Safe Mode. As an additional set of steps, you can first try running a full on your system to clear caches and other odd temporary files before reapplying the update.
If applying the update will not fix the issue, you might be experiencing conflicts with other filesystems. Try unmounting external storage devices and Boot Camp partitions to see if that allows the backup to continue. If so then try remounting the drives to see if the next backup run (invoke it manually from the system menu bar) continues without problems. If you do not want to back up a specific secondary filesystem and Time Machine works without it, you can add the filesystem to the exclusion list that's available if you click the "Options" button in the Time Machine system preferences.
Lastly, one issue that might result in pauses and hangs with Time Machine is if the Time Machine disk is not functioning correctly, either because of hardware issues or because of filesystem structure corruption. Be sure to check all of your drives (including the Time Machine drive) with Disk Utility or a more robust option like DiskWarrior, Drive Genius, TechTool Pro, or DiskTools Pro, and fix any problems that crop up.
If the drive is experiencing hardware issues, you might be able to see these in the Console utility by selecting the "Console Messages" or "All Messages" sections. Read, write, or I/O errors with the drive may be output to the console in these sections. In addition to the console, you can try using one of the disk utilities or a specialized SMART reporter (SMARTReporter or SMART Utility) to check the SMART status of the drive (internal drive diagnostics). While not always a reliable source for determining drive problems, SMART may provide some information about hardware malfunctions. Keep in mind that SMART is only available through a local drive controller, and cannot be read through FireWire, USB, or networking protocols. If a drive is giving hardware-related errors, it is likely best to consider replacing the drive.