Issue: Normally it only takes a few minutes for Time Machine to complete its hourly backup, but now it seems like it's taking forever. I've waited N minutes before stopping it (where N > a few minutes), tried rebooting, cycling power on my Time Capsule/external backup disk, etc, but it still gets stuck on “Preparing Backup...”.
Solution: Fortunately, the solution to this problem is usually... patience. Certain events can cause Time Machine to lose track of which files it needs to backup, which necessitates performing a full comparison of your current hard drive contents to the contents of the last backup. This can cause the backup preparation stage to last for several tens of minutes to more than an hour, depending on the size of the drive you are backing up.
Events that can cause this problem include:
(a) the last Time Machine backup was interrupted (other than by issuing the Stop Backing Up command from the Time Machine toolbar menu),
(b) your Mac was not shut down properly,
(c) a mounted disk was not removed properly by ejecting it first,
(d) a software update was installed (the OS X 10.5.2 Leopard update was notorious for causing this problem), and
(e) your computer was used for several days without performing a Time Machine backup (e.g., by not having the Time Machine disk connected or having backups turned off in the Time Machine preferences).
You can confirm that this is the origin of your problem with Time Machine as follows: after Time Machine starts (while it is “Preparing Backup”), open Console (from the Applications → Utilities folder). Near the bottom of the system log file in Console, there should be one or more time-tagged entries relating to the ongoing Time Machine process. If one of these mentions a (somewhat ominously named) “deep traversal” or “deep scan”, then just be patient, let Time Machine do its thing, and all will be well in the future.
Update added on 04 Feb 2010: Apple claims that Time Machine is “up to 80 percent faster” in OS X 10.6 Snow Leopard. So, this problem might not exist anymore. I haven't run into it on my Snow Leopard machines.
Update added on 28 Oct 2012: Although I haven't personally run into this issue in years (!), it appears to still be present as a possible Time Machine behavior under OS X 10.8 Mountain Lion; cf. http://support.apple.com/kb/PH11239.
Update Status: 28 Oct 2012